Length versus width

[TinyTree]

This is an attempt to approach the probability of the universe from a new angle (pun intended).

If we look at the Cartesian coordinate system, a two dimensional rectangular coordinate system, we can imagine a series of connected line segments forming a figure.

Now a friend may come and ask "What is the length of that figure?"

You may immediately respond "It is five units in length", and you are measuring from the left hand side of the figure to the right hand side of the figure, along the x axis. Your measurement is completely accurate, and you can verify it again and again.

This is similar to asking "what is the probability of a physical system?"

There is an established method of measuring probability- which considers the probability of entering all microstates which correspond to a phase space of a partition, which is a mutually disjoint and exhaustive set of all states the physical system can reside within. This is known as the entropy of the system and yields the thermodynamic probabilities for said system.

But if you look at the figure some more, you may notice that you could also measure it along the y axis. Instead of measuring from left to right along the x axis, you could measure the height. And in fact, if you measure the height you can see that it is 3 units long. So now you might tell your friend "It is 3 units long"

This is similar to looking at the probability of a specific microstate- the probability of the state vector. There is an alternate measure of the probability of a physical system. It is related to the entropy of the system in that it is also a measure of the probability of the physical system. It does not negate, or make the entropy measure incomplete. It is a different measure of the probability of the system.

Your friend would reply "What is it- is it 5 units long like you just told me, or is it 3 units long? Which is it? Is it 3 or 5 units in size?"

What would your response be?

It is of course both 5 units along the x axis and 3 units along the y axis. The original problem was that your friend asked "What is the length of that figure?" anticipating you would tell him one number. A single number does not encompasse the two lengths you could suggest.

This is the same as asking "What is the probability of a physical system?" The answer is that there is not a single number to encompasse this measurement. There are two numbers: the entropy of the system, and the probability of the state vector.

So let us look at some questions about the probability of the state vector with regards to this analogy.

One reader said "Statistical mechanics already includes the probability of quantum mechanics." This is similar to saying in the measurement of the diagram "The measurement from the lower bound on the x axis to the higher bound of the x axis already includes the length of the figure." This is completely true- but it does not mean you can't measure it along the other axis.

Thus you can see that the probability of the state vector does not suggest that all of existing statistical mechanics is wrong. It does not suggest that quantum mechanics is left out of statistical mechanics. It is an alternative measurement of the probability of a physical system.

By the way, "probability of the state vector" is an unusually long phrase for a physical quality, and it is in fact confusing because it is implicitly confused with entropy. I would like a new word for the probability of the state vector, that could be slipped in its place, similar to the word "Entropy" in that it is just a couple syllables, but different.

If anyone has a good suggestion for this measurement, please let me know what it is. It is as if we have a word for width, but not one for height.

[Guille]

I'm still thinking for a name for the probability of a state vector.

But actually I do think you can give a two number answer: matrices. Actually, vectors, for it would only be two numbers. If we get into cartesian coordinate then y is the porbability of the state vector and x is the probability by entropy, then a point in the plane will tell you the porbability of the system which the point is representing in both dimensions. You should jsut write it as a vecotr, with one column and two rows; the top is x and the bottom is y. The good thing of this idea is that it makes it possible to thus calculate the probability of the state vector and of entropy of a collection of system: you can do a line, which can be curved, and thus you could be connecting calculus to probability, you can study areas, as groups of systems (for ex: the probability of all humans is an area in the plane)....

[TinyTree]

I'm still thinking for a name for the probability of a state vector.

But actually I do think you can give a two number answer: matrices. Actually, vectors, for it would only be two numbers. If we get into cartesian coordinate then y is the porbability of the state vector and x is the probability by entropy, then a point in the plane will tell you the porbability of the system which the point is representing in both dimensions. You should jsut write it as a vecotr, with one column and two rows; the top is x and the bottom is y. The good thing of this idea is that it makes it possible to thus calculate the probability of the state vector and of entropy of a collection of system: you can do a line, which can be curved, and thus you could be connecting calculus to probability, you can study areas, as groups of systems (for ex: the probability of all humans is an area in the plane)....

This is a very interesting idea. Plot the phase space of a system with a two coordinate axis representing both the entropy and the probability of the state vector. When I get a chance to do some more mathematical analysis of the state vector of some mathematical systems this would be interesting to see.

The only slight difficulty is that normally one would speak of the probability of the state vector changing through time, so one would show a graph with time on one axis, and the probability of the state vector on the other. Also, one would normally show the entropy of a system changing through time- so it is a graph of time vs entropy.

Thus, a robust graph would have three dimensions- one for time, one for probability of the state vector, and one for entropy. There are tools for drawing three dimensional graphs and this would be interesting to see.

I like the idea of showing phase space as a volume of probabilities- that is- some sort of concrete physical phenomena represented as a set of probabilities. IE- some sort of self propagating low probability structure (like an organism) represented out in the phase space as all future possible instances of that organism.

[Guille]

Maybe a 3 dimensional graph is neccesary. But maybe not. What if we add to the dimensions x and y the time already to each? I mean, by this, that the x axis is the probability of a state vector per time and the y axis is the probability of entropy per time. Thus, each dimension would actually be measuring the intensity (sort of) instead of the probabilities themselves.

I think actually histograms are perfect for this.

[mkirkpatrick]

what is the length of forever,and how wide is eternity!
What is it you are trying to do measure the universe!!
it would be easier to weigh it,would it not.




kindest regards michael.

[Guille]

Mike,

No. It would be much easier to measure the size of the universe than it's mass. First, because even if it's impossible to measure the size the automatically it is impossible to measure the mass. And second, because the mass depends on gravity, but, the gravity is different in everyplace, whiles the space is equal everywhere (I took this last thing from Quanta07's theory. In opposition with Mike5's theory).

[mkirkpatrick]

How high is the sky!I suppose you could answer that if you were to qualify that
with as far as the last molcule that is detected,you couldthen give a fairly accurate reply.But you would have a very difficult task if you tried to see how
high was the universe,and the question that arises from this is that if the universe is eternal,then any measurement would be futile and absurd.Length and
width amount to zero when confronted with the eternal,whether a thing be infinitely large,or infinitely small on the backdrop of the eternal,they are equal.



kind regards michael.

[Guille]

Mike,

Yes, if the universe is infinite (no end in space) or eternal (no end in time):
would be impossible to measure it's spatio-temporal occupation. But also it
would be impossible to measure it's mass.
If spacetime has no end, then neither does mass.
And you can't measure it's mass, for if it's eternal you will have to be adding eternally, and if it's infinite the same.
And the exact reason why I am quasi-absolutelly and quasi-completely convinced 99.99999999999999999999999999999% that the universe is FINITE in the spatio-temporal-mass meaning, is because we can have measurements of subgroups of these, thus, quantities exist, thus, there is no supra-quantity existence.
This means, I can measure my mass, my length, width and thickness, and also how much time I have occupied, and, I'm finite, this impplies that the system I'm in is finite.

[mkirkpatrick]

we can and do measure things,it makes sense to have a shirt,or a pair of jeans
that fit you,I can fully understand that,and If I am having my house built again
I would want it measured out,no problem.But when it comes to the universe,then that is a different story.If you have no starting point and no end in sight,then your measurments are useless!The universe to all intents and purposes is eternal,no known beginning,and no known ending,plenty of theories
but no actual facts.It is like asking how long is a piece of string?who knows,and who really cares!This universe is NOT finite my friend,it is an eternal cycle of
expansion and contraction,and there is no ruler on earth to measure it.And even
if you could what then would you do with this result,accept that there is a greater intelligence than ourselves.Or maybe try and find the square root of it!

kind regards michael.

[Guille]

Michael,

Is there any scientific or at least philosophical proof that the universe is eternal/infinite? NO. And actually, the observations done talk oppositelly to this proposition: everything we have come up to is finite, therefore the probability that we come up to an infinite thing, either if it's the universe, or the macrouniverse composed of universes, is zero, 0, nothing, nothingness, absence, null, non-existing. I know induction is crap, but we are humans, so we are crap, therefore the best methods we can use to understand the universe, are also crap, such as induction. We should stay to TinyTree's theme.