# Thread: Is sphere is an the only absolute symmetry shape?

1. Well, the title isn't what I actually wanted to ask. However, I believe that sphere is the only absolute symmetry shape in infinite axis which passes through its center.

What I want to know is, say we have a absolute sphere ball which is consist of an fully filled with air which the ball cannot expand anymore, as further pumping air will cause the ball explode.

As a few air inside is leaked, the ball will not shrink since its membrane is inelastic, but causing the ball become a different shape and will not became symmetry anymore.

Is this statement is true? Will an absolute symmetry inelastic ball will no more become absolute symmetry again after a little air is leaked? (The ball might be in symmetry in a few axis but still not a absolute symmetric way.)

I am thinking of this question relating to the broken symmetry in particle physics. I am thinking of this "air leaking phenomena" which might related to symmetry breaking.

2.

3. Since you are trying to think about it in terms of particle physics (no idea how you got there in the first place), then nobody would know exactly what you mean, since it is not mainstream physics.

In terms of a normal ball, there would be other things to consider, like what the outside air pressure is for instance, how flexible the ball material is, etc.

4. I am just a hobbyist about physics especially the issue regarding the unsolved physics mystery on broken symmetry. I am kind of very hard to explain my thought here since the broken symmetry is not simple because it related to various matter.

Ok, I will try to make the question simple. First, I am thinking of the starting of the big bang, were everything is super-symmetry and it should be like a sphere. (I understand that the 'thing' here cannot just be a sphere because it needed be related to imagery dimensions, however I just assumed that is should be like this first.)

At first I 'found' that is very difficult to 'imagine' as well to draw how the sphere from absolute symmetry transform into non-symmetry. But one thing is I realize that the a ball of leaking air has no symmetry at all!

Maybe taking a leaking ball is not a good example. So lets change the situation where there is a absolute symmetrical sphere energy ball, at which it is the first energy form of all universe, having its spherical 'membrane' (it is not a REAL membrane, just the shape of the world at that time) which means the edge of the universe, which content the energy inside. I think of the logic that if the energy ball is just expanding (or contracting, meaning the expansion-contraction is sort of an oscillation), will not cause the spherical 'membrane' to transform into non-symmetrical way. But 'a leak' will cause the energy become non-symmetry.

I am very hard to describe the 'leak' in proper any either.

So the question I am asking is, kind of like this: We have a spherical ball with its surface area fixed. As the air volume inside the ball is enough, the ball will 'looked' spherical, then it will be a total symmetry right? However, as a few air volume is leaked outside, the ball will become... leaked. (I done have any word to describe that) So, the ball is now will not be any symmetrical again.

Is this statement is true?

5. Looking at symmetry in physics is not as simple as looking at a sphere. In particle physics it's not a matter of geometric symmetry, physicists look at things such as charge symmetry, parity symmetry, and time symmetry.

6. Originally Posted by lol1986
I am just a hobbyist about physics especially the issue regarding the unsolved physics mystery on broken symmetry. I am kind of very hard to explain my thought here since the broken symmetry is not simple because it related to various matter.

Ok, I will try to make the question simple. First, I am thinking of the starting of the big bang, were everything is super-symmetry and it should be like a sphere. (I understand that the 'thing' here cannot just be a sphere because it needed be related to imagery dimensions, however I just assumed that is should be like this first.)

At first I 'found' that is very difficult to 'imagine' as well to draw how the sphere from absolute symmetry transform into non-symmetry. But one thing is I realize that the a ball of leaking air has no symmetry at all!

Maybe taking a leaking ball is not a good example. So lets change the situation where there is a absolute symmetrical sphere energy ball, at which it is the first energy form of all universe, having its spherical 'membrane' (it is not a REAL membrane, just the shape of the world at that time) which means the edge of the universe, which content the energy inside. I think of the logic that if the energy ball is just expanding (or contracting, meaning the expansion-contraction is sort of an oscillation), will not cause the spherical 'membrane' to transform into non-symmetrical way. But 'a leak' will cause the energy become non-symmetry.

I am very hard to describe the 'leak' in proper any either.

So the question I am asking is, kind of like this: We have a spherical ball with its surface area fixed. As the air volume inside the ball is enough, the ball will 'looked' spherical, then it will be a total symmetry right? However, as a few air volume is leaked outside, the ball will become... leaked. (I done have any word to describe that) So, the ball is now will not be any symmetrical again.

Is this statement is true?
As chemboy noted, the idea of symmetry at the level of physics to which you are alluding is not quite so simple as looking at a sphere. If you really want to chase such problems you need to learn some fairly deep physics and mathematics. The symmetries involve the actions of symmetry groups, in this case certain Lie groups, on the spaces on which the fundamental equations are defined. This subject is rather advanced and is far beyond what we can address quickly in a forum such as this. You would really need to read a book on quantum field theory, and have the necessary background in math and physics to understand what you are reading.

7. Originally Posted by lol1986
Well, the title isn't what I actually wanted to ask. However, I believe that sphere is the only absolute symmetry shape in infinite axis which passes through its center.
Well this three dimensional degenerate continuous rotational symmetry certainly defines the sphere in the context of a being embedded in R^3 (infinite 3 dimensional continuous space).

Originally Posted by lol1986
What I want to know is, say we have a absolute sphere ball which is consist of an fully filled with air which the ball cannot expand anymore, as further pumping air will cause the ball explode.

As a few air inside is leaked, the ball will not shrink since its membrane is inelastic, but causing the ball become a different shape and will not became symmetry anymore.

Is this statement is true? Will an absolute symmetry inelastic ball will no more become absolute symmetry again after a little air is leaked? (The ball might be in symmetry in a few axis but still not a absolute symmetric way.)

I am thinking of this question relating to the broken symmetry in particle physics. I am thinking of this "air leaking phenomena" which might related to symmetry breaking.
Yes even without any external symmetry breaking force like earth gravity it will spontaneously break the symmetry in an unpredictable way and this is indeed the same idea as spontaneous symmetry breaking in particle physics. What you have to understand however that the scope and type of symmetries in particle physics is considerable greater than what is found in a sphere.

8. Time, parity, charge, and how many symmetries are there that had been 'discovered'?

What about spin, baryon, lepton and kinds of quantum number, are they considered as symmetries also?

One more thing, is it possible, or it should be, that all these symmetries are actually come out from one single symmetry? If it should be, what will be the single symmetry is/was?

9. Originally Posted by lol1986
Time, parity, charge, and how many symmetries are there that had been 'discovered'?

What about spin, baryon, lepton and kinds of quantum number, are they considered as symmetries also?
Yes every conservation law is associated with some type of symmetry because in quantum field theory the symmetry can be used to mathematically derive the associated conservation law. This is called Noether's theorem. I believe that even broken symmetries can be said to be associated with weak conservation principles.

Originally Posted by lol1986
One more thing, is it possible, or it should be, that all these symmetries are actually come out from one single symmetry? If it should be, what will be the single symmetry is/was?
That doesn't really make any sense the way you have said it. What we would say is that a bunch of different symmetries can be said to be a part of a higher symmetry. When you look up the old string theories you will in fact see the different ones labeled with a symbolic mathematical representation for the type of higher symmetry which it is based upon.

10. Originally Posted by mitchellmckain
Originally Posted by lol1986
One more thing, is it possible, or it should be, that all these symmetries are actually come out from one single symmetry? If it should be, what will be the single symmetry is/was?
That doesn't really make any sense the way you have said it. What we would say is that a bunch of different symmetries can be said to be a part of a higher symmetry. When you look up the old string theories you will in fact see the different ones labeled with a symbolic mathematical representation for the type of higher symmetry which it is based upon.
So you mean that the symmetries shouldn't be coming from one unity symmetry? But then how come that symmetries will occur at the beginning of the universe before it knows how to interact themselves? Moreover, how can the symmetries produced another higher new symmetries?

Or how about that the fundamental symmetries are fixed at the beginning? Meaning the symmetries are always there, fixed, and there would never new symmetries come out from our universe?

11. Originally Posted by lol1986
Originally Posted by mitchellmckain
Originally Posted by lol1986
One more thing, is it possible, or it should be, that all these symmetries are actually come out from one single symmetry? If it should be, what will be the single symmetry is/was?
That doesn't really make any sense the way you have said it. What we would say is that a bunch of different symmetries can be said to be a part of a higher symmetry. When you look up the old string theories you will in fact see the different ones labeled with a symbolic mathematical representation for the type of higher symmetry which it is based upon.
So you mean that the symmetries shouldn't be coming from one unity symmetry? But then how come that symmetries will occur at the beginning of the universe before it knows how to interact themselves? Moreover, how can the symmetries produced another higher new symmetries?

Or how about that the fundamental symmetries are fixed at the beginning? Meaning the symmetries are always there, fixed, and there would never new symmetries come out from our universe?
Well I am not sure I understand what you are asking, but the idea is that the universe began with some higher symmetry which was then spontaneously broken leaving remnants of that original symmetry unbroken. Other parts of the symmetry also became broken in a manner similar to your ball example, only instead of leaking air, it is the decreasing temperature of the universe as it expands which is the cause. This is one way that they verify the theory -- by recreating the high temperature conditions of the early universe in the particle accelerators, and thus restoring some of that original symmetry.

12. Originally Posted by mitchellmckain
Well I am not sure I understand what you are asking, but the idea is that the universe began with some higher symmetry which was then spontaneously broken leaving remnants of that original symmetry unbroken. Other parts of the symmetry also became broken in a manner similar to your ball example, only instead of leaking air, it is the decreasing temperature of the universe as it expands which is the cause. This is one way that they verify the theory -- by recreating the high temperature conditions of the early universe in the particle accelerators, and thus restoring some of that original symmetry.
I think I had a misunderstanding here. I thought that the symmetries are broking from lower to higher manner. I think that's why you don't understand, because I make mistake at there already.

***

Yea from lab results it showed that at higher temperature the symmetries 'seem' to be converge together. But for me it doesn't make sense, because the higher temperature cause the entropy to become chaotic. What I mean here ('entropic chaos') is that higher temperature is irreversible from thermal 2nd law stated and I felt the symmetries should be diverge at that manner rather than restoring symmetries.

...or there is something I missed to understand. For me from 1st generation to 3rd generation of fundamental particles observed, it seems to be diverging rather than converging, at both symmetries and particles.

***

I think that that we don't know wither at the beginning there should be highest temperature state because we don't have observable prove, so do the nothingness of universe. But when I think about that which the initial energy that conserved stilled and create the universe now is not logic. I rather believe that the nothingness energy that creates the universe.

And that is the part of my idea for what I am asking about this sphere symmetry breaking, that the breaking of one unity symmetry is the key that creates the whole universe. It might seem to be contradict the current science, but as I thinking the logic that pushing and pulling how should the universe beginning for and beginning from, this should be correct.

But I still leave myself open minded for what will be the facts are.

***

Ok, if we have initial higher symmetries fixed at the beginning is there any theory or logic that support or explain that? How should they coming from? For me this question is seems to be answer-less, because I didn't see any science theory that available yet for this.

13. Originally Posted by lol1986
Ok, if we have initial higher symmetries fixed at the beginning is there any theory or logic that support or explain that? How should they coming from? For me this question is seems to be answer-less, because I didn't see any science theory that available yet for this.
Science is a ground up methodology. We look at what is and try to explain that. So the justification for whatever higher symmetry the universe originally had is is found in the fact that it makes the theory work to explain what we see and measure.

However in Modern string theory (i.e. M-theory) all these different string theories with their different dimensions and symmetry groups have been shown to be aspects of a super-theory. I am not quite sure what that means in regards to these symmetries. It probably means that these are all parts of a higher symmetry of some kind, but the construction of M-theory was very different from the way that these string theories were constructed and so I am not sure if this is the case. It is possible that there is some sort of arbitrariness or relativity in regards to the overall symmetry of the universe in the sense that this symmetry depends on how you look at it or on how you give it a mathematical formulation.

14. Originally Posted by mitchellmckain
However in Modern string theory (i.e. M-theory) all these different string theories with their different dimensions and symmetry groups have been shown to be aspects of a super-theory. I am not quite sure what that means in regards to these symmetries. It probably means that these are all parts of a higher symmetry of some kind, but the construction of M-theory was very different from the way that these string theories were constructed and so I am not sure if this is the case. It is possible that there is some sort of arbitrariness or relativity in regards to the overall symmetry of the universe in the sense that this symmetry depends on how you look at it or on how you give it a mathematical formulation.
I don't really like string theory because for me it still didn't prove anything yet but comes out new theory that unproven yet though I not really understand it. String? Doesn't make any sense to me.

However, I found a not famous theory that done by Burkhard Heim and I think this one has far better than the string, however I barely understand his theory too. What I notice in his work is first he didn't putting or having our generally known particle physics ideas such as symmetry breaking, gauge field, higgs boson, quantum entanglement and others. Second, his works are united by matrices that can derive a lot of particle physics things like masses of each fundamental particles and their half-lifes, which are very impressing me tough I don't understand it. However this theory is lacking of explaining the particles' interactions. I wish I can understand heim theory well so that I could find the answer I wanted.

If heim's theory is right, then our questions to particle physics can be solved faster by understand it. Unfortunately this theory is very unpopular.

15. Originally Posted by lol1986
If heim's theory is right, then our questions to particle physics can be solved faster by understand it. Unfortunately this theory is very unpopular.
Well don't be fooled by this word popular to think that science is some kind of popularity contest. Nothing could be farther from the truth. Instead what it is really all about is which theory provides people in science opportunities for conducting research. So saying that a theory is "unpopular" is just a way of saying that no one has found any way to do any further research in that direction an thus as far as we know it is a dead end. It is possible that this could change and it does sometimes happen, BUT (and this is a really big BUT) it is usually because the reasearch in a different direction to open up possibilites which makes this dead end a fruitful direction for research. And that invariably means that some kind of synthesis is involved, for it is difficult to say which theory has enabled research in which theory, that is, it kinda has to go both ways.

In the case of string theory however it is not just that this is providing active avenues of profitable research but that what it is promising to explain really exceeds that of any other competing theory, so much so that it is most likely the case, that any involvement of a theory like the one you speak of will be percieved as enabling the research into string theory to continue, rather than the other way around.

16. Originally Posted by lol1986
Originally Posted by mitchellmckain
However in Modern string theory (i.e. M-theory) all these different string theories with their different dimensions and symmetry groups have been shown to be aspects of a super-theory. I am not quite sure what that means in regards to these symmetries. It probably means that these are all parts of a higher symmetry of some kind, but the construction of M-theory was very different from the way that these string theories were constructed and so I am not sure if this is the case. It is possible that there is some sort of arbitrariness or relativity in regards to the overall symmetry of the universe in the sense that this symmetry depends on how you look at it or on how you give it a mathematical formulation.
I don't really like string theory because for me it still didn't prove anything yet but comes out new theory that unproven yet though I not really understand it. String? Doesn't make any sense to me.

However, I found a not famous theory that done by Burkhard Heim and I think this one has far better than the string, however I barely understand his theory too. What I notice in his work is first he didn't putting or having our generally known particle physics ideas such as symmetry breaking, gauge field, higgs boson, quantum entanglement and others. Second, his works are united by matrices that can derive a lot of particle physics things like masses of each fundamental particles and their half-lifes, which are very impressing me tough I don't understand it. However this theory is lacking of explaining the particles' interactions. I wish I can understand heim theory well so that I could find the answer I wanted.

If heim's theory is right, then our questions to particle physics can be solved faster by understand it. Unfortunately this theory is very unpopular.
Heim theory may be unpopular for a reason. In any case the planned near-term experiments at the LHC may confirm the existence of the Higgs boson, which would apparently put the nails in the coffin of Heim theory. Or perhaps a non-detection would renew interest. http://en.wikipedia.org/wiki/Heim_theory

17. [quote="mitchellmckain"]
Originally Posted by lol1986
...
However in Modern string theory (i.e. M-theory) all these different string theories with their different dimensions and symmetry groups have been shown to be aspects of a super-theory. ...
Are you sure of that statement ? If so, do you have a reference.

My understanding is that this is a conjecture of Ed Witten that is sometimes stated as a theorem but is in fact not proven. If a proof has been constructed it would be good to have a reference.

18. Originally Posted by mitchellmckain
In the case of string theory however it is not just that this is providing active avenues of profitable research but that what it is promising to explain really exceeds that of any other competing theory, so much so that it is most likely the case, that any involvement of a theory like the one you speak of will be percieved as enabling the research into string theory to continue, rather than the other way around.
Yeah I know what you mean.

In fact that almost all western science research is involved with money, where one research needed financial support to help it going further.

However particle physics seems to be different because human spent a lot of money into but coming out result bit by bit and is mostly improve technology itself but didn't come out with much technology that improve human life. Compared to wildlife exploration documentary, particle physics didn't give much profit because high cost involved. I think this is the weirdest happening in science world ever since.

Correct me if I am wrong.

19. You know, I bet many people thought that Edison's or Tesla's experiments and inventions seemed pretty pointless at the time too, but hindsight is 20/20. Come back in 20-50 years and tell me if particles physics research is really useless.

20. One thing we got from particle physics is we now have a very open sighted and creativity of learning about of our world. Fictions is one of the best product' from it. Another changes brought is a lot of people seems to relating particle physics' theories with religion and philosophy. I didn't say this is worthless, but it mean to be like that.

The puzzles of particles physics had been more than 50 years already, and it is 50 years after death of Einstein, what had we got from it??

21. Well, from general relativity, among other things, we have the GPS system and we have a much more precise understanding of the history of the universe. But maybe I'm underestimating the time it takes for basic research to pay off. The point is that it will. It nearly always has before.

22. Originally Posted by MagiMaster
Well, from general relativity, among other things, we have the GPS system and we have a much more precise understanding of the history of the universe. But maybe I'm underestimating the time it takes for basic research to pay off. The point is that it will. It nearly always has before.
I am sorry but that is not completely accurate. It is true that some of the first attempts used SR&GR calculations to correct for these effect but then I think later a better way (using reference signals perhaps) was found to do this that doesn't use such calculations at all.

I have to agree with lol1986 that the technological benefits from many areas of physics research like particle physics has been scarce for a while and this is why there is less and less money for it, especially in a comercial driven country like the U.S.A.

On the other hand, it pays to know as much about things as you can because you never know when your survival might depend on it.

Originally Posted by DrRocket
Originally Posted by mitchellmckain
However in Modern string theory (i.e. M-theory) all these different string theories with their different dimensions and symmetry groups have been shown to be aspects of a super-theory. ...
Are you sure of that statement ? If so, do you have a reference.

My understanding is that this is a conjecture of Ed Witten that is sometimes stated as a theorem but is in fact not proven.
Your question does not make a whole lot of sense to me. My statement is about M-theory and that is what M-theory is whether there is any proof for M-theory or not. I'm sure I have no idea of what theorem of Ed Witten's you might be talking about. There certainly has been considerable more work on M-theory since 1995 and what I said has certainly been the presumption of the scientific for some time now. As for a reference, you could take me as reference or you can use google and either you will find something that satisfies you or not. Or you could make more of an effort to explain what you mean and give us a quote of this theorem you are talking about.

23. I've never heard of them doing anything to get rid of the need for such corrections, and I have recently heard (on TV though, so I don't know how long ago it was recorded) physicists saying that GPS was a good example of the uses of relativity.

24. Originally Posted by MagiMaster
I've never heard of them doing anything to get rid of the need for such corrections, and I have recently heard (on TV though, so I don't know how long ago it was recorded) physicists saying that GPS was a good example of the uses of relativity.
Well yes it remains an example of that despite the fact that they do not actually use it any more. But the real point is that it certainly rather forcefully verified the truth of the theory didn't it?

Besides there are a lot of satelites put up there and maybe some of them still use the relativity calculation method.

25. Posts in this thread of a questionable nature and responding posts have been moved to:
http://www.thescienceforum.com/Is-sphere-is-an-the-only-absolute-symmetry-shape(pseudo)-15645t.php

Some posts may have been edited to remove references to the moved posts.

Any posts attempting to continue the discussions of the moved posts in this thread shall be deleted.
Janus

26. Originally Posted by mitchellmckain

Originally Posted by DrRocket
Originally Posted by mitchellmckain
However in Modern string theory (i.e. M-theory) all these different string theories with their different dimensions and symmetry groups have been shown to be aspects of a super-theory. ...
Are you sure of that statement ? If so, do you have a reference.

My understanding is that this is a conjecture of Ed Witten that is sometimes stated as a theorem but is in fact not proven.
Your question does not make a whole lot of sense to me. My statement is about M-theory and that is what M-theory is whether there is any proof for M-theory or not. I'm sure I have no idea of what theorem of Ed Witten's you might be talking about. There certainly has been considerable more work on M-theory since 1995 and what I said has certainly been the presumption of the scientific for some time now. As for a reference, you could take me as reference or you can use google and either you will find something that satisfies you or not. Or you could make more of an effort to explain what you mean and give us a quote of this theorem you are talking about.
Prior to 1995 there were 5 competing string theories under study -- Type I, Type IIA, Type IIB, Heterotic-O and Heterotic-E. In 1995 Ed Witten gave a talk and wrote a paper http://arxiv.org/PS_cache/hep-th/pdf/9503/9503124v2.pdf which is sometimes quoted as providing a dictinary among those theories and thereby showing that they are actually just different presentations of a single theory -- that theory being M theory. However, it is my understanding that in fact Witten did not provide a proof of this unification, but rather outlined reasons for believing that it ought to be true -- so he formulated a conjecture rather than a theorem. So far as I know this theorem has not been proved. I was asksing if you knew of a proof and had a reference. So far as I know none exists, although many physicists proceed as thought it does exist.

Wiki is not necessarily up to date, but this article indicates that the theorem proving unification has not yet been proved. http://en.wikipedia.org/wiki/M-theory

27. Originally Posted by MagiMaster
You know, I bet many people thought that Edison's or Tesla's experiments and inventions seemed pretty pointless at the time too, but hindsight is 20/20. Come back in 20-50 years and tell me if particles physics research is really useless.
No the people loved the idea of Tesla, and his inventions and experiments, they also knew that he would have to use the devices as weapons, on Washington DC before anyone could enjoy them. We still do not have those devices for all.

Sincerely,

William McCormick

28. Originally Posted by DrRocket
Originally Posted by mitchellmckain
Your question does not make a whole lot of sense to me. My statement is about M-theory and that is what M-theory is whether there is any proof for M-theory or not. I'm sure I have no idea of what theorem of Ed Witten's you might be talking about. There certainly has been considerable more work on M-theory since 1995 and what I said has certainly been the presumption of the scientific for some time now. As for a reference, you could take me as reference or you can use google and either you will find something that satisfies you or not. Or you could make more of an effort to explain what you mean and give us a quote of this theorem you are talking about.
Prior to 1995 there were 5 competing string theories under study -- Type I, Type IIA, Type IIB, Heterotic-O and Heterotic-E. In 1995 Ed Witten gave a talk and wrote a paper http://arxiv.org/PS_cache/hep-th/pdf/9503/9503124v2.pdf which is sometimes quoted as providing a dictinary among those theories and thereby showing that they are actually just different presentations of a single theory -- that theory being M theory. However, it is my understanding that in fact Witten did not provide a proof of this unification, but rather outlined reasons for believing that it ought to be true -- so he formulated a conjecture rather than a theorem. So far as I know this theorem has not been proved. I was asksing if you knew of a proof and had a reference. So far as I know none exists, although many physicists proceed as thought it does exist.

Wiki is not necessarily up to date, but this article indicates that the theorem proving unification has not yet been proved. http://en.wikipedia.org/wiki/M-theory
Neither of the links you provided mention any such theorem

But in any case, I think we can indeed agree that M-theory is not any more proven to be correct than any of these string theories that it supposedly "unites". But even if you are correct, I do not think it is to correct to say that physicists proceed as though this proof does exist but simply that they see M-theory as a fruitful direction for research. Let us not forget that not only is there no proof that mathematics is consistent but we actually have a proof that such a proof is impossible. So we might as well admit that science does not wait until an approach it is proven to be legitimate. Often the proof is in pudding so to speak: in this case that means that the direction of scientific research is led by the finding of things to investigate and explore - led by curiosity and ambition perhaps we could say? It is difficult, but we must be careful to distinguish between the possibilities that scientific research is investigating and what it actually knows to be true.

29. Originally Posted by mitchellmckain
Originally Posted by DrRocket
Originally Posted by mitchellmckain
Your question does not make a whole lot of sense to me. My statement is about M-theory and that is what M-theory is whether there is any proof for M-theory or not. I'm sure I have no idea of what theorem of Ed Witten's you might be talking about. There certainly has been considerable more work on M-theory since 1995 and what I said has certainly been the presumption of the scientific for some time now. As for a reference, you could take me as reference or you can use google and either you will find something that satisfies you or not. Or you could make more of an effort to explain what you mean and give us a quote of this theorem you are talking about.
Prior to 1995 there were 5 competing string theories under study -- Type I, Type IIA, Type IIB, Heterotic-O and Heterotic-E. In 1995 Ed Witten gave a talk and wrote a paper http://arxiv.org/PS_cache/hep-th/pdf/9503/9503124v2.pdf which is sometimes quoted as providing a dictinary among those theories and thereby showing that they are actually just different presentations of a single theory -- that theory being M theory. However, it is my understanding that in fact Witten did not provide a proof of this unification, but rather outlined reasons for believing that it ought to be true -- so he formulated a conjecture rather than a theorem. So far as I know this theorem has not been proved. I was asksing if you knew of a proof and had a reference. So far as I know none exists, although many physicists proceed as thought it does exist.

Wiki is not necessarily up to date, but this article indicates that the theorem proving unification has not yet been proved. http://en.wikipedia.org/wiki/M-theory
Neither of the links you provided mention any such theorem

But in any case, I think we can indeed agree that M-theory is not any more proven to be correct than any of these string theories that it supposedly "unites". But even if you are correct, I do not think it is to correct to say that physicists proceed as though this proof does exist but simply that they see M-theory as a fruitful direction for research. Let us not forget that not only is there no proof that mathematics is consistent but we actually have a proof that such a proof is impossible. So we might as well admit that science does not wait until an approach it is proven to be legitimate. Often the proof is in pudding so to speak: in this case that means that the direction of scientific research is led by the finding of things to investigate and explore - led by curiosity and ambition perhaps we could say? It is difficult, but we must be careful to distinguish between the possibilities that scientific research is investigating and what it actually knows to be true.
In general I would agree. But string theory and M-theory are not yet physics and the research is more nearly mathematics than physics.

The question here is not whether M-theory is internally consistent, but whether M-theory is definable. You are correct, neither reference mentions a theorem. That is the point. I can find to reference to such a theorem actually having been proved. The paper of Witten's that was linked was the paper that introduced M-theory. He suggested that M-theory would be a unification of the 5 theories in existence at the time (that is essentially a definition of M-theory), and provided some ratinale as to why such a unificatin is plausible. But he did not actually provide the proof that there exists such a unification. So the path that one might expect would be to first produce the "dictionary" linking the 5 theories and then work out the consequences of that unified theory. But so far as I know the "dictionary" has not been completed and M-theory remains conjectural. Thus the first big question for M-theory remains "What is M-theory". That in particular prevents it from really being a physical theory that can make predictions that then would be put to the experimentalists for verification.

I agree that once you have a predictive physical theory the proof is in the pudding. But until you have a model capable of prediction you cannot even start to mix the pudding, let along put it in the oven.

I think it is quite fair to say that physicists are proceeding as though the linkage between the 5 earlier theories that yields M-theory has been proven. That is not to imply that proceeding in that manner is wrong, but it is to say that there is a logical gap that will need to be filled at sometime. And it is to say that the statement that M-theory unites other theories is more a definition of what an M-theory would be if it exists than a statement that such a unification has been demonstrated. 'That is, unless the linkage has in fact been proven and it is simply the case that neither one of us know of it.

Godel's incompleteness theorem with respect to the ability to prove the internal consistency of the foundations of mathematics (say for instance Zermelo Frankel set theory or any set theory that permits construction of the real numbers) is not relevant to this question.

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