Register to reply

If the universe is infinite, does that mean that everything exists somewhere?

by Richard87
Tags: exists, infinite, universe
Share this thread:
Chalnoth
#127
Feb9-11, 04:37 AM
Sci Advisor
P: 4,804
Quote Quote by A. Neumaier View Post
Can you show why it should follow from that???
If the integration over macrostates is limited at some high energy, and every component of that integration is finite (that is, if the function is well-defined everywhere), then it will have to be finite, because it will be a representation of a sum over a finite (but large) number of states.

Quote Quote by A. Neumaier View Post
If you look at the books of statistical mechanics, you'll find that microstates means only ''energy eigenstate'', and not ''arbitrary state''.
I don't think this is true at all. The basis you do your sums in is completely irrelevant. It has to be, by nature of the underlying mathematics. The only reason why the sums are done in the energy basis is because:
1. Most introductory statistical mechanics books neglect complications like spin, angular momentum, and any other potential quantum numbers that are different from energy.
2. It is much easier to do the sums in terms of energy because the total energy of the system is one of the macroscopic variables we use.

In principle you could always change to some other basis, and if it's done right you have to come up with the exact same answer, but it's going to be much more difficult to connect the other basis to the macroscopic variables.

That said, this is an off-topic argument, because it simply has no application to my original statement, which had nothing whatsoever to do with entropy. Remember, I was making two separate points when talking about the finite number of potential states. The entropy argument was one argument, and is a separate one from the purely quantum-mechanical one.

The purely quantum-mechanical argument is that as long as you cut off your states at some high energy, there are a finite (though large) number of states. You came back and stated that you can also have superpositions of those states, and since there can be an infinite number of superpositions, this finite number of quantum states leads to an infinite number of possibilities.

Not so, I said, because the superpositions are merely a representational issue: any superposition of states can be represented as an eigenstate of the right operator. You'll still always get the exact same number of states, no matter the representation you use, as long as you do the counting correctly. This second argument for the finite number of states has nothing to do with statistical mechanics.
A. Neumaier
#128
Feb9-11, 04:53 AM
Sci Advisor
P: 1,943
Quote Quote by Chalnoth View Post
If the integration over macrostates is limited at some high energy, and every component of that integration is finite (that is, if the function is well-defined everywhere), then it will have to be finite, because it will be a representation of a sum over a finite (but large) number of states.
But this can be argued only locally. The energy cutoff of QFT is something at the level of individual scattering events, while the integration over macrostates in statistical mechanics never had such a cutoff.


Quote Quote by Chalnoth View Post
I don't think this is true at all. The basis you do your sums in is completely irrelevant. It has to be, by nature of the underlying mathematics. The only reason why the sums are done in the energy basis is because:
No. The only reason why the sums are done in the energy basis is because the canonical ensemble involves the Hamiltonian, and the trace defining the entropy reduces to a sum _only_ in a representation where the basis states are energy eigenstates.


Quote Quote by Chalnoth View Post
The purely quantum-mechanical argument is that as long as you cut off your states at some high energy, there are a finite (though large) number of states.
And I pointed out that both your hypothesis and your conclusion are flawed.
Chalnoth
#129
Feb9-11, 05:07 AM
Sci Advisor
P: 4,804
Quote Quote by A. Neumaier View Post
But this can be argued only locally. The energy cutoff of QFT is something at the level of individual scattering events, while the integration over macrostates in statistical mechanics never had such a cutoff.
Typically you don't do any integration over macrostates in statistical mechanics. The integrations are over microstates. And you don't need any cutoff there because we are generally considering systems that are at such low temperatures that any cutoff that would come in from high-energy physics is irrelevant.

But when considering all possible states of the system, you have to integrate the number of states over the ensemble of all possible macrostates. As long as the number of states for any given macrostate is finite, and as long as you have to cut off your integral at some energy (so that the integral doesn't go to infinite), the result also has to be finite.

Quote Quote by A. Neumaier View Post
No. The only reason why the sums are done in the energy basis is because the canonical ensemble involves the Hamiltonian, and the trace defining the entropy reduces to a sum _only_ in a representation where the basis states are energy eigenstates.
And the reason why the canonical ensemble includes the Hamiltonian is because energy is one of the macroscopic variables. It is the only operator used because in the classical treatment, energy is the only thing that is allowed to be mixed (the particle number and volume tend to be fixed). When considering more complicated systems, such as a quantum system including spin or one where the particle number is allowed to vary, you have to make the ensemble a bit more complicated, so that it incorporates these added degrees of freedom.

It doesn't really matter, though. You can still transform to another basis if you like. The results will necessarily come out the same. It's just that the math will be horribly difficult, and thus it's much easier to just remain in the eigenbasis of your ensemble.

Quote Quote by A. Neumaier View Post
And I pointed out that both your hypothesis and your conclusion are flawed.
No, because you changed topics and started talking about statistical mechanics in an argument that had nothing to do with statistical mechanics.
A. Neumaier
#130
Feb9-11, 05:47 AM
Sci Advisor
P: 1,943
Quote Quote by Chalnoth View Post
No, because you changed topics and started talking about statistical mechanics in an argument that had nothing to do with statistical mechanics.
As if entropy and counting quantum states could be done without statistical mechanics.
Chalnoth
#131
Feb9-11, 05:50 AM
Sci Advisor
P: 4,804
Quote Quote by A. Neumaier View Post
As if entropy and counting quantum states could be done without statistical mechanics.
Huh? Counting states is a component of statistical mechanics, but hardly requires it. Entropy doesn't even need to come into the argument when all you're interested in is the total number of possible states.
Terraqueous
#132
Feb9-11, 01:50 PM
P: 1
Quote Quote by Entropee View Post
About how long did it take for the quark-gluon plasma to cool?
I'm not completely sure on this, but I think the answer is 10^-6 seconds.
GODISMYSHADOW
#133
Feb9-11, 02:48 PM
P: 40
Quote Quote by A. Neumaier View Post
This is undecidable.

Suppose you'd record every detail about the history of the universe, wait till it has died, and then replay it in a perfect simulation (where of course, everything is already there). The physical laws would be exactly the same - without the slightest detectable difference.
Are you suggesting the universe is a simulation?
A. Neumaier
#134
Feb10-11, 02:21 AM
Sci Advisor
P: 1,943
Quote Quote by GODISMYSHADOW View Post
Are you suggesting the universe is a simulation?
No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect.
Tanelorn
#135
Feb10-11, 11:21 AM
Tanelorn's Avatar
P: 728
"No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect."


I am not sure I understand what you are actually saying there but I can say that there is a great difference between a mathematical simulation on a computer with a cpu executing single arithmetic instructions one bit at a time and the space time reality we are part of. In a similar way it is highly unlikely that life like intelligence can ever be created on such a simple calculating device.
A. Neumaier
#136
Feb10-11, 11:44 AM
Sci Advisor
P: 1,943
Quote Quote by Tanelorn View Post
"No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect."


I am not sure I understand what you are actually saying there but I can say that there is a great difference between a mathematical simulation on a computer with a cpu executing single arithmetic instructions one bit at a time and the space time reality we are part of. In a similar way it is highly unlikely that life like intelligence can ever be created on such a simple calculating device.
Of course. If our universe were a simulation, it would have been simulated on one of God's hyper-computers with a very different physics and technology.

The point is, we couldn't see the difference in the results.
Tanelorn
#137
Feb10-11, 11:58 AM
Tanelorn's Avatar
P: 728
Or the Universe and God could be one and the same thing. No simulation required :)
GODISMYSHADOW
#138
Feb10-11, 02:19 PM
P: 40
Quote Quote by A. Neumaier View Post
Of course. If our universe were a simulation, it would have been simulated on one of God's hyper-computers with a very different physics and technology.

The point is, we couldn't see the difference in the results.
So many different views! To me, the universe is a probability with no
provable objective reality.
Tanelorn
#139
Feb18-11, 08:54 AM
Tanelorn's Avatar
P: 728
Quote Quote by Tanelorn View Post
Or the Universe and God could be one and the same thing. No simulation required :)


After hearing Anthony Hopkins discuss his support yesterday of the Philosopher Spinoza's views I decided to dig a little deeper and was pleasantly surprised that I share many of the sentiments:


Albert Einstein named Spinoza as the philosopher who exerted the most influence on his world view (Weltanschauung). Spinoza equated God (infinite substance) with Nature, consistent with Einstein's belief in an impersonal deity. In 1929, Einstein was asked in a telegram by Rabbi Herbert S. Goldstein whether he believed in God. Einstein responded by telegram: "I believe in Spinoza's God who reveals himself in the orderly harmony of what exists, not in a God who concerns himself with the fates and actions of human beings." Spinoza's pantheism has also influenced environmental theory; Arne Nss, the father of the deep ecology movement, acknowledged Spinoza as an important inspiration.


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


I apologise if this is overly philosophical, I will not add to this, I just thought it was an interesting comment.
Chalnoth
#140
Feb18-11, 10:02 AM
Sci Advisor
P: 4,804
Yeah, I personally never liked that idea as it always seemed to me that "God" carried with it far too much anthropomorphic meaning to be anything but misunderstood when used in that way. It sounds like an attempt to re-purpose the religious word to describe some feeling of awe or wonder regarding the universe itself. But I just don't see the purpose in doing that. Can't we describe the majesty of the universe without resorting to anthropomorphic words? And there remains, to me, a significant downside in that the religious merely use it as an excuse to trumpet their own views (the religious absolutely love to imagine that science is on their side, and famous scientists talking about "God" are exceptionally tantalizing).
Tanelorn
#141
Feb19-11, 08:34 AM
Tanelorn's Avatar
P: 728
Chalnoth, I sympathise with your views also. In fact I find I can move between Atheism, Agnosticism and Pantheism, sometimes all on the same day. Perhaps in his statement Einstein was helping by leading people from the old superstition anthropomorphic based religions into a higher state of enlightment, taking baby steps so to speak. Hopefully we will avoid the fate that Sagan was so concerned about. The main reason I have for sometimes believing in something greater is that it sometimes appears to me that there was a very powerful and intentional force behind the creation of the universe. It cant be proven, but the universe seems so finely tuned, too much so for random chance. The whole thing seems so unlikely, and instead we could have had a universe consisting of nothing more than an infinite amount of green jelly!

In the Anthony Hopkins interview, a fellow Welshman, I particularly agreed with his views regarding people "who know the truth". Such certainties gave rise to people like Hitler with plans for everyone. I have come to similar conclusions myself.
Deuterium2H
#142
May20-11, 02:13 PM
P: 59
Quote Quote by Sage Lee View Post
For me, the biggest surprise was that an empty set has a cardinality of 1. (Did I say this right?) This just pissed me off, and got me reading about vacuous truth, and it wasn't long before I threw my hands up in exasperation and stopped trying to understand why.

But because of my frustration, I didn't like the joke "in a set of zero mathematicians, any one of them can do it [change a light bulb]." I actually remarked, to no one in particular, that "in a set of zero mathematicians, three of them are actually tomatoes." I liked this better because, "Hey, if we're being ridiculous, let's just let it all hang out and be ridiculous." What can I say, I was annoyed and was on that previously described tomato kick at the time.

But whatever, I accept on faith alone that an empty set is actually "one," because Wikipedia told me so... but I don't have to like it.

But all in all, I really, really like Set Theory, because as I said, with it, it seems possible to describe just about anything at all using math.


Congratulations, to both you and Chalnoth. I now completely agree with that statement. Gold star for youse guys. Although I'm thinking, as I said before, that I never really disagreed, I just didn't understand what infinity actually meant (I thought it literally meant "exhaustive.")
Hi Sage,
Sorry for resurrecting this older thread, but I happened to be re-reading through it for another reason, and had previously missed a statement you made, in error, that may cause all sorts of confusion if left uncorrected. The Cardinality of the Empty Set (Null Set) is not one, it is zero. The Set that contains the Empty Set is equal in Cardinality to one. In fact, in axiomatic Set Theory (e.g. ZFC), the existence of the Empty Set is defined as fundamental Axiom. It is upon this, and the following Pair Set and Sum Set axioms that larger Sets are created...thusly:

{ }= = 0
{{ }} = {} = 1
{{{ }}} = {,{}} = {0,1} = 2
{{{{ }}}} = {,{},{,{}}} = {0,1,2} = 3
etc., etc.
nakian
#143
Jul19-11, 07:18 AM
P: 2
I read some answers that tended to argue that the possibility that everything could exist was unlikely. Other comments gave the impression that having a twin in another world sounded like sci-fi... Maybe you should spend some times reading what Max Tengmark has to say about the Multiverse http://space.mit.edu/home/tegmark/PD...erse_sciam.pdf. Also find out more here http://en.wikipedia.org/wiki/Multive...ogical_horizon.

The argument Tengmark makes is that worlds similar to ours are very likely, that is the likelihood that you have a twin somewhere in another world is high. Those un-observable universes , those of level-I, that is worlds beyond our cosmological horizon, will probably be of an infinite number. They will all have the same physical laws and constants as ours. Everything that is possible in our world will be possible in those worlds. In that sense, everyhing that could happen here, even if it will never happen here or has never happened here, would probably have happened or will probably happen somewhere in a Level-I un-observable world. In conclusion, it is highly probable that you have a twin somewhere, dating J-Lo's twin in that world...
nakian
#144
Jul19-11, 07:50 AM
P: 2
Quote Quote by Tanelorn View Post
Albert Einstein named Spinoza as the philosopher who exerted the most influence on his world view (Weltanschauung). Spinoza equated God (infinite substance) with Nature, consistent with Einstein's belief in an impersonal deity. interesting comment.
By impersonal do we mean "unconscious"? Because, Spinoza God also possesses the Attribute of being infinitely conscious. I am not to sure what being infinitely conscious means, but I pretty sure it's not the same thing as "unconscious". Am I confusing things here?


Register to reply

Related Discussions
The Universe - infinite or not ? Cosmology 139
Will humans ever really understand why the universe exists? General Discussion 254
Infinite-infinite universe explains all General Discussion 42
Does an infinite universe effect an infinite outlook? Astronomy & Astrophysics 11