Quantum Physics Question

[ark]

Here I see that some little mathematics may be needed. The following rules as to how operate with "infinity" are taken from "Probability Theory", Vol I, by M. Loeve:

In these rules x is a finite number.

You will find there in particular that ∞+5=∞ and that ∞-∞ is meaningless (indeterminate, can be any number you wish).

 

[Approaching Infinity]

If the universe is an open system, then that means there's a possibility of adding energy into the system. Even if the Universe operates without any input for a long time (giving us the impression that it is closed, from our perspective), that doesn't preclude the possibility of an increase of energy. And if the energy is infinite, ∞+5=∞, then we would'nt be able to tell if energy had been added. So one could "prove" to oneself that the system is closed! Or so it seems to me...

 

[Possibility of Being]

Now, what about this piece:

The First Law of Thermodynamics - energy can be neither created nor destroyed - is one of the most profound laws of existence. Few people understand its full significance. It means that no new energy can ever be created and no existing energy can ever disappear. Once energy exists it exists forever, and since energy cannot be created, all the energy that exists now must always have existed. No new energy will ever be arriving, and none will ever be departing. In other words, the energy of the universe is the same now as it was a trillion years ago....

The First Law of Thermodynamics works for closed and isolated systems only as far as I remember. In the above that person takes for granted that the universe is such a system. Maybe it is and maybe it isn't, I don't know. Anyway, I smell a hidden hypothesis here.

is the same now as it was a trillion years ago, and the same as it will be a trillion years from now on, and the same as it will be at any other time we care to choose. Energy can only ever be transformed, never made or unmade.

At least, that's one interpretation. There are actually two ways to view the energy content of the universe according to the First Law. Either it is always ZERO i.e. there is always precisely enough "negative" energy (from gravitation) to exactly counterbalance "positive" energy (from particles), resulting in a total universal energy of zero...

One type of energy can be transform in the other one, right? So at a given moment the distribution of energy changes. But that person says the total sum always stays zero. How so? Where does the energy needed to counterbalance that change come from (the system is isolated)? In a case of infinite energies (positive and negative) and finite changes at any given moment there is no point in trying to calculate the difference.

or the amount of energy in the universe is always INFINITE.

Why?

Science has never been able to definitively prove which option is correct. Any other value of the universe's total energy is absurd since there would never be a sufficient reason for the energy content of the universe to be an arbitrary amount. (After all, why would one amount be any more likely than any other amount?)

What would be the sufficient reason for the energy content to be zero or infinity? Isn't ZERO or infinity an arbitrary amount the same way any finite number would be? Why would zero be any more likely than any other amount?

 

[ark]

Now, what about this apparent alternative: "Either it is always ZERO i.e. there is always precisely enough "negative" energy (from gravitation) to exactly counterbalance "positive" energy (from particles), resulting in a total universal energy of zero...or the amount of energy in the universe is always INFINITE".

What if it is indeterminate? What if there is a "quantum uncertainty" about the amount of energy in the Universe? What if our Universe is one of the many, that somehow, "once in a while" or "continuously" are able to interact one with another? You will certainly be able to think of more scenarios.

But talking about the Universe is always based on assumptions, and you can at most speculate one way or another. You can't really deduce any truth from such speculations, because these assumptions (or beliefs) are impossible to verify.

These are at least my thoughts...

 

[Buddy]

I guess I am just having trouble wraping my head around what infinite energy could possibly mean. I can visualize a scenario with zero energy or almost zero energy like a satellaite in its proper orbit to circle around the earth for a very long time without having to expend energy to stay there. But infinite energy? What the heck does that mean?

I like the example of the satellite in orbit, it's something I can wrap my mind around. A satellite has potential energy, due to its altitude, with a force vector pointing inward to the center of the earth, down the gravity well. It also has centripetal force, in the opposite direction, due to its rotational velocity around the Earth, holding it in place, altitude-wise. If somehow you could magically make that satellite indestructible, put it in orbit around an object with a theoretically infinite gravity well, like say a black hole, and give it infinite velocity, so its orbit will maintain due to the infinite centripetal force outward from the black hole, you have three infinities, all different directions, but the net expenditure is zero.

WhiteBear, thanks for that post. I was reminded of the idea of 'potential infinity' which I can relate to. Also, the example of the gravity well reminded me of the pathological infinite regress version of 'infinity' where losing touch with the spirit of a discussion can lead you into finally questioning the meaning of 'a' and 'the'.

Now, as for that math...the only way I can think of it at the moment is "forever + 5". Otherwise an 'infinite' number of something would require an 'infinite' space to hold it and I (as observer) would probably get 'pushed out of the system' and lose count. :)

 

[Guardian]

Here I see that some little mathematics may be needed. The following rules as to how operate with "infinity" are taken from "Probability Theory", Vol I, by M. Loeve:

How do Physicists come up with "probabilities" regarding things like energy, matter, etc. when they don't even know a TINY fraction of the possibilities? Infinite concepts, finite minds while in meatsacks...Helloooo?

 

[Laura]

How do Physicists come up with "probabilities" regarding things like energy, matter, etc. when they don't even know a TINY fraction of the possibilities? Infinite concepts, finite minds while in meatsacks...Helloooo?

Well, at least they are trying. At least some of them are. And you can't just stand around and curse the darkness for being so intense - if you only have a candle, at least that's a candle in the darkness!

 

[lamalamalamalama]

Here I see that some little mathematics may be needed. The following rules as to how operate with "infinity" are taken from "Probability Theory", Vol I, by M. Loeve:

In these rules x is a finite number.

You will find there in particular that ∞+5=∞ and that ∞-∞ is meaningless (indeterminate, can be any number you wish).

So any thoughts on using surreal numbers to examine these 'problems' of infinities in physics?

http://en.wikipedia.org/wiki/Surreal_number

 

[StrangeCaptain]

Here I see that some little mathematics may be needed. The following rules as to how operate with "infinity" are taken from "Probability Theory", Vol I, by M. Loeve:

How do Physicists come up with "probabilities" regarding things like energy, matter, etc. when they don't even know a TINY fraction of the possibilities? Infinite concepts, finite minds while in meatsacks...Helloooo?

This is where math comes in. It's a language, and the proper use of it can help the mind to shine some light on difficult topics just as the use of written language, Laura's books for example, can propel the mind to think about a topic in a new way.

Infinity is difficult for the mind to grasp, but with fairly basic math and logic, one can come to the inescapable conclusion that there are infinities of different size. The mind may still have problems grasping infinity, but with the proper application of the language of math, one can see how the infinity of the counting numbers (1,2,3,...) is "smaller" than the infinity of the real numbers (those numbers like 1/4 and 1/2 where we can always find another number between any two). Further we can sense the truth through this language that the infinity of the counting numbers is the SAME as the even counting numbers (2,4,6,...). In fact, there are the same number of even counting numbers as there are counting numbers. This is not intuitive but a proper use of the language of math can convince us that this might very well be true. But don't worry if this seems weird... The mathematicians of the 1800'a and early 1900's were haveing the same doubts when they were trying to flesh out calculus. Newton had crazy intuition and trying to record in logical detail how he came to some of his conclusions was quite an issue for many years.

Anyway, it is late. I just got off of work, but tomorrow I hope this discussion continues because it brings up some questions I have about my own studies that I probably can formulate more meaningfully after some serious sleep. Thanks all.

 

[Guardian]

Well, at least they are trying. At least some of them are. And you can't just stand around and curse the darkness for being so intense - if you only have a candle, at least that's a candle in the darkness!

You're right of course...everyone needs to do their own thing, in their own way. I probably shouldn't have even hopped into this topic...mentioning a known predator in a thread is like waving a bunch of feathers on a string in front of my cat. I just gotta bat at it a bit

Sorry if I intruded and got snarky.

 

[WhiteBear]

Infinity is difficult for the mind to grasp, but with fairly basic math and logic, one can come to the inescapable conclusion that there are infinities of different size. The mind may still have problems grasping infinity, but with the proper application of the language of math, one can see how the infinity of the counting numbers (1,2,3,...) is "smaller" than the infinity of the real numbers (those numbers like 1/4 and 1/2 where we can always find another number between any two).

I'm curious...is this true? In my mind's eye, I'm seeing the progression of counting numbers stretching out of sight, and the progression of real numbers, fractional steps becoming smaller and smaller, (the old "half the distance to the wall" bet) stretching out of sight, but into a microscopic level, and I'm not seeing a difference in the "size" of them. With the first, you never reach the last number, and with the second, you never reach zero...(or the wall) :)

 

[John G]

So any thoughts on using surreal numbers to examine these 'problems' of infinities in physics?

Well I would say the infinities of surreal numbers might be used to explore things in physics that aren't necessarily infinity problems in physics:

http://www.valdostamuseum.org/hamsmith/surreal.html

 

[John G]

Infinity is difficult for the mind to grasp, but with fairly basic math and logic, one can come to the inescapable conclusion that there are infinities of different size. The mind may still have problems grasping infinity, but with the proper application of the language of math, one can see how the infinity of the counting numbers (1,2,3,...) is "smaller" than the infinity of the real numbers (those numbers like 1/4 and 1/2 where we can always find another number between any two).

I'm curious...is this true? In my mind's eye, I'm seeing the progression of counting numbers stretching out of sight, and the progression of real numbers, fractional steps becoming smaller and smaller, (the old "half the distance to the wall" bet) stretching out of sight, but into a microscopic level, and I'm not seeing a difference in the "size" of them. With the first, you never reach the last number, and with the second, you never reach zero...(or the wall) :)

I think maybe for real numbers Patience means to include not only numbers between 0 and 1 but between 1 and 2, 2 and 3, etc. so that it counts up to out of sight too but includes the numbers in between the counting numbers.

 

[John G]

Well, at least they are trying. At least some of them are. And you can't just stand around and curse the darkness for being so intense - if you only have a candle, at least that's a candle in the darkness!

You're right of course...everyone needs to do their own thing, in their own way. I probably shouldn't have even hopped into this topic...mentioning a known predator in a thread is like waving a bunch of feathers on a string in front of my cat. I just gotta bat at it a bit

Sorry if I intruded and got snarky.

This was a topic with kind of a don't believe everything you read message but it was also a little fun too (at least for me). Historically physicists have certainly done some impressive things. Einstein's relativity, quantum mechanics, quarks, things like these have the advantage of experiments to confirm the interesting ideas. For the universe there's even an experiment that gives ratios for how much exotic stuff like dark energy and dark matter there is compared with regular matter. There is a good starting point.

In this forum, the idea of adding in the possibility of communicating with the future, doing time travel, makes sense as something to add in to the good starting point. Where mainstream physics (string theory) runs into trouble is adding a bunch of stuff that is far away from the good starting point.

 

[combsbt]

Infinity is difficult for the mind to grasp, but with fairly basic math and logic, one can come to the inescapable conclusion that there are infinities of different size. The mind may still have problems grasping infinity, but with the proper application of the language of math, one can see how the infinity of the counting numbers (1,2,3,...) is "smaller" than the infinity of the real numbers (those numbers like 1/4 and 1/2 where we can always find another number between any two).

I'm curious...is this true? In my mind's eye, I'm seeing the progression of counting numbers stretching out of sight, and the progression of real numbers, fractional steps becoming smaller and smaller, (the old "half the distance to the wall" bet) stretching out of sight, but into a microscopic level, and I'm not seeing a difference in the "size" of them. With the first, you never reach the last number, and with the second, you never reach zero...(or the wall) :)

I think maybe for real numbers Patience means to include not only numbers between 0 and 1 but between 1 and 2, 2 and 3, etc. so that it counts up to out of sight too but includes the numbers in between the counting numbers.

The difference between countable sets and uncountable sets has to do with their cardinality.

Case 1: | A | = | B |

Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.

For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E.

[edit] Case 2: | A | ≥ | B |

A has cardinality greater than or equal to the cardinality of B if there exists an injective function from B into A.

[edit] Case 3: | A | > | B |

A has cardinality strictly greater than the cardinality of B if there is an injective function, but no bijective function, from B to A.

For example, the set R of all real numbers has cardinality strictly greater than the cardinality of the set N of all natural numbers, because the inclusion map i : N → R is injective, but it can be shown that there does not exist a bijective function from N to R (see Cantor's diagonal argument).

And here's Cantor's diagonal argument.

It is possible to build a sequence of elements s0 in such a way that its first element is different from the first element of the first sequence in the list, its second element is different from the second element of the second sequence in the list, and, in general, its nth element is different from the nth element of the nth sequence in the list. That is to say, if sm,m is 1, then s0,m is 0, otherwise s0,m is 1. For instance:

s1 = (0, 0, 0, 0, 0, 0, 0, ...)
s2 = (1, 1, 1, 1, 1, 1, 1, ...)
s3 = (0, 1, 0, 1, 0, 1, 0, ...)
s4 = (1, 0, 1, 0, 1, 0, 1, ...)
s5 = (1, 1, 0, 1, 0, 1, 1, ...)
s6 = (0, 0, 1, 1, 0, 1, 1, ...)
s7 = (1, 0, 0, 0, 1, 0, 0, ...)
...

s0 = (1, 0, 1, 1, 1, 0, 1, ...)

(The elements s1,1, s2,2, s3,3, and so on, are here highlighted, showing the origin of the name "diagonal argument". Note that the highlighted elements in s0 are in every case different from the highlighted elements in the table above it.)

Therefore it may be seen that this new sequence s0 is distinct from all the sequences in the list. This follows from the fact that if it were identical to, say, the 10th sequence in the list, then we would have s0,10 = s10,10. In general, if it appeared as the nth sequence on the list, we would have s0,n = sn,n, which, due to the construction of s0, is impossible.

From this it follows that the set T, consisting of all infinite sequences of zeros and ones, cannot be put into a denumerable list s1, s2, s3, ... Otherwise, it would be possible by the above process to construct a sequence s0 which would both be in T (because it is a sequence of 0s and 1s which is by the definition of T in T) and at the same time not in T (because we can deliberately construct it not to be in the list). T, containing all such sequences, must contain s0, which is just such a sequence. But since s0 does not appear anywhere on the list, T cannot contain s0.

Therefore T cannot be placed in one-to-one correspondence with the natural numbers. In other words, it is uncountable.

The interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable. In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.

Sorry for just quoting wikipedia but I think that's a fairly easy to follow explanation. For more information on different uncountable sets and their relative "size" see _http://en.wikipedia.org/wiki/Cardinality_of_the_continuum