How can Gravity have particles.

[Juan R.]

Hmm Connes uses (implicitly) the concept of convergence, as topology relies on it. And yes, convergence uses the concept of infinite sequences, but you can recast it in the infamous epsilon/delta format, enough for operative purposes. The concept of infinitesimal as used in calculus is a bit more sophisticated that the concept of limit, the subtle points about differential calculus come because it involves the simultaneous use of various infinite/infinitesimal quantities. Eg $$\delta x \over \delta t$$ or $$\sum^\infty_i \delta x_i$$.

It is very hard to get rid of the concept of convergence. You lose a lot of math, for instance the numbers pi and e. But you can admit convergence and still be suspicius about simultaneus use of infinit* entites; in this sense it is very welcome that quantum mechanics is against a one of these simultanities: position and momentum. Remember also that primitive quantum mechanics was formulated as an index theorem (the bohr/sommerfeld formulation) so it is very interesting that Connes abstraction drives also to this kind of theorems.

No problem with concept of convergence for series n but Connes takes the eigenvalue of the infinitesimal explicitly from n --> infinite on the operator.

Note that he does n --> infinite, cannot do n = infinite. Precisely, this indicates the same inconsistency that classical definition using limits.

 

[arivero]

Connes takes the eigenvalue of the infinitesimal explicitly from n --> infinite on the operator.

Explicitly? Never seen it. In some lecture notes Connes explicitly builds a compact operator as an infinite-dimensional matrix, you could be mistaken because of it. There is no such a thing as "the eigenvalue of the operator". In Connes the infinitesimal is the whole operator, no an eigenvalue from it, nor a limit in the eigenvalues.

The equivalent in Connes of the two infinite processes is the interplay between the definition of a compact operator and the action of Dixmier trace upon it.

 

[Juan R.]

Explicitly? Never seen it. In some lecture notes Connes explicitly builds a compact operator as an infinite-dimensional matrix, you could be mistaken because of it. There is no such a thing as "the eigenvalue of the operator". In Connes the infinitesimal is the whole operator, no an eigenvalue from it, nor a limit in the eigenvalues.

The equivalent in Connes of the two infinite processes is the interplay between the definition of a compact operator and the action of Dixmier trace upon it.

Let me rewrite it

Connes define the infinitesimal like the limit n --> infinite of the mu(n) wher mu(n) is the characteristic value of operator. He uses explicitely

n --> infinite

He cannot write n = infinite. This is the same inconsistency that traditional math has.

 

[arivero]

Connes define the infinitesimal like the limit n --> infinite of the mu(n) wher mu(n) is the characteristic value of operator.

Nego. Show me that definition, reference and page number please.

 

[Juan R.]

Nego. Show me that definition, reference and page number please.

arXiv:math.QA/0011193 v1 23 Nov 2000, pag 22.

He uses explicitely n --> infinite for the infinitesimal of order alpha.

The problem of correctly understanding

n --> infinite

arises in Connes approach.

The nonstandard approach says that n cannot be a real or complex number, it is a hiperreal.

Are I wrong about this?

 

[arivero]

arXiv:math.QA/0011193 v1 23 Nov 2000, pag 22.

He uses explicitely n --> infinite for the infinitesimal of order alpha.

The problem of correctly understanding

n --> infinite

arises in Connes approach.

Ok I see. Look two parragraphs below formula (1)

Since the size of an infinitesimal is measured by the sequence $$\mu_n \to 0$$... A variable would just be a bounded sequence, and an infinitesimal a sequence $$\mu_n, \mu_n \to 0$$.

(Bold emphasis mine). And just the parragraw above (1).

The size of the infinitesimal $$T \in K$$ is governed by the order of decay of the sequence of characteristic values $$\mu_n=\mu_n(T)$$ as $$n\to\infty$$.

Here, to define $$\mu_n$$ out of $$T$$ is where I believe we have used the Axiom of Choice (via Zorn's Lemma) as I commented above, but perhaps the condition in parenthesis pass formula (1) is able even to cincunvent this.

The paper proceeds to explain why it is interesting to consider all these sequences as sequences of eigenvalues of an operator, in order to go beyond the commutative algebra. It makes intereresting reading, but every line asks for deep mathematical understanding.

The nonstandard approach says that n cannot be a real or complex number, it is a hiperreal.

Are I wrong about this?

The $$n\in *R$$ in Robinson is as you say an hyperreal, but this is unrelated to the discussion on Connes paper, where $$n\to \infty$$ is simply the traditional notation to indicate the convergence conditions of an infinite series (which you can interpret with the traditional $$\forall \epsilon \exists \delta...$$)

 

[Kea]

A single coordinate in a noncommutative space has a continuum spectrum, that is the reason why non-commutative geometry is not a substitute for quantization of spacetime. LQG quantizes spacetime including single coordinates (lengths).

Hi guys

Firstly, it is quite possible to discuss continuum lengths in the context of spin foams. For example, using the representation category of a non-compact group such as the Poincare group introduces a continuous parameter label for unitary representations which may be used in the 2-categorical labelling of 4D triangulations.

Having said this, I believe that one of the best arguments against standard NCG is that an a priori control on length values in QG is simply not good enough. To quote Connes from the abstract of the paper Juan mentions: The basic tools of the theory: K-theory, cyclic cohomology, Morita equivalence, operator theoretic index theorems, Hopf algebra symmetry...

At least three of these items is more than incidentally category theoretic, and yet when it comes to hard analysis this aspect of NCG seems to be ignored.
On page 22 Connes dismisses non-standard analysis on the grounds that it is non-constructible, saying a non-standard number is some sort of chimera ... but here he is definitely missing the point. He only references a book from 1969 which predates an incredible amount of work on constructive logic and constructive analysis etc. We now understand what it means to operate within a logic and Robinson's topos, far from being non-constructive, is the best logic with which to discuss infinitesimals.

Cheers
Kea

 

[Kea]

P.S. Connes 4 page motivational introduction, on quantum physics, is wonderful...

http://arxiv.org/abs/math/0011193

 

[arivero]

On page 22 Connes dismisses non-standard analysis on the grounds that it is non-constructible, saying a non-standard number is some sort of chimera ... but here he is definitely missing the point. He only references a book from 1969 which predates an incredible amount of work on constructive logic and constructive analysis etc. We now understand what it means to operate within a logic and Robinson's topos, far from being non-constructive, is the best logic with which to discuss infinitesimals.

Kea, yes, I also feel Connes could be missing some of the possibilites there. Particullarly I wonder about how the ultrafilter/ultraproducts construction of Robinson reals is dismissed, but the point about topoi seems sensible too.

 

[Juan R.]

Ok I see. Look two parragraphs below formula (1)

Since the size of an infinitesimal is measured by the sequence $$\mu_n \to 0$$... A variable would just be a bounded sequence, and an infinitesimal a sequence $$\mu_n, \mu_n \to 0$$.

(Bold emphasis mine). And just the parragraw above (1).

The size of the infinitesimal $$T \in K$$ is governed by the order of decay of the sequence of characteristic values $$\mu_n=\mu_n(T)$$ as $$n\to\infty$$.

Here, to define $$\mu_n$$ out of $$T$$ is where I believe we have used the Axiom of Choice (via Zorn's Lemma) as I commented above, but perhaps the condition in parenthesis pass formula (1) is able even to cincunvent this.

The paper proceeds to explain why it is interesting to consider all these sequences as sequences of eigenvalues of an operator, in order to go beyond the commutative algebra. It makes intereresting reading, but every line asks for deep mathematical understanding.



The $$n\in *R$$ in Robinson is as you say an hyperreal, but this is unrelated to the discussion on Connes paper, where $$n\to \infty$$ is simply the traditional notation to indicate the convergence conditions of an infinite series (which you can interpret with the traditional $$\forall \epsilon \exists \delta...$$)

Exactly! Connes is using an infinitesimal in the usual delta-epsilon definition for defining other infinitesimal. He is using the "-->" which is not rigorously defined in standard math. Therein the born of nonstandard analysis. Those problems are not present in epsilon calculus and, yes, some of initial motivations regarding hiperreals by Connes are wrong.

However, Connes main objetive is not substitute nonstandard analisys, it is, i believe, to quantize geometry using tools from quantum mechanics and i think, Kea, that there it has failed.

It is true that NC is being studied in generalizations of string brane theory but LQG has really quantized space and time. In all programs of quantum gravity i think that NC geometry is less sucesfull of all.

 

[arivero]

All we are telling is that a sequence goes to zero as the secuence progresses. Usual notion of convergence of sequences, unrelated to the notion coming from Robison infinitesimals. It goes to zero. To zero. Not to any infinitesimal number. It goes to zero as the sequence progresses. Progresses. Not reaching infinity, just progressing from i to i+1 indefinitely ever and ever. You can from some philosophical standpoints to say that the sequence is wrong defined because it never ends. It is a valid philosophic aptitude, and in this way you deny the notion of convergence of a series, independently of the number the series is converging to. In Connes's case, series are converging to a very well known finite number, namely zero. You told me in a previous message that you were happy with the notion of convergence, but now it seems you confuse this notion with the one of infinitesimal and then you attack it. If you do not admit convergence, we can not proceed anymore in this mathematical way, and I am not sure if there is any other.

What Connes is telling there is that for instance the sequence

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,... (convergent to zero)

defines an infinitesimal of different size (order) that the sequence, for instance,

1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49,... (convergent to zero).

The only concepts involved are series and convergence of series.

Additionaly, Connes assumes that it is possible to extract a ordered series out of the eigenvalues of an operator. As I told, this could be resting on Zorn's Lemma if the series is to be extracted via an orthonormal basis, but I am not sure of the degree of generality required.

But I can not see any circularity in the arguments. I say more: there is no circularity in the argument. The infinitesimals defined by Connes are not used in the definition of convergence as you seem to believe in your uncareful reading. They are used against (in duality with?) Dixmier trace to build a theory of differentials and integration.

If you do not believe convergence, eg if you do not admit that classical mathematics can stablish that 1+1/2+1/4+1/8+1/16+1/32+1/64 --> 2 or that the decimal numbers 1.999999999999... and 2.0000000000000... are the same real number, then of course you can not admit Connes's aproach. I entered in this discussion mistaken by your previous affirmation about not having problems with convergence.

Dejame añadir, por estar seguro de que se me sigue, que las dos series que pongo en este mensaje son ejemplos exactos (no metaforas ni nada por el estilo) del tipo de series que Connes esta describiendo en los parrafos que he citado. Las series no usan ningun infinitesimal ni convergen a un infinitesimal (de hecho convergen a cero). Las series *son* los infinitesimales.

 

[Kea]

Connes main objective is not substitute nonstandard analysis, it is, I believe, to quantize geometry using tools from quantum mechanics and I think, Kea, that there it has failed.

It depends what you mean by failure. One can hardly say that the tools that Connes has developed are trivial and useless. On the contrary, they define genuine non-commutative spaces, many of which appear in quantum physics. If, however, you are interested in QG spaces, then I tend to agree...but these shouldn't be referred to as quantum as such, because the arguments are necessarily post-quantum. Someone really needs to invent a new word for this.

 

[Juan R.]

It depends what you mean by failure. One can hardly say that the tools that Connes has developed are trivial and useless. On the contrary, they define genuine non-commutative spaces, many of which appear in quantum physics. If, however, you are interested in QG spaces, then I tend to agree...but these shouldn't be referred to as quantum as such, because the arguments are necessarily post-quantum. Someone really needs to invent a new word for this.

I agree.

By failure i mean initial Connes objectives of deriving SM from purely geometrical issues. I also mean that NCG is not very sucesfull in quantum gravity. In fact, i think that is one of less sucessful programs in quantum gravity today.

 

[Juan R.]

All we are telling is that a sequence goes to zero as the secuence progresses. Usual notion of convergence of sequences, unrelated to the notion coming from Robison infinitesimals. It goes to zero. To zero. Not to any infinitesimal number. It goes to zero as the sequence progresses. Progresses. Not reaching infinity, just progressing from i to i+1 indefinitely ever and ever. You can from some philosophical standpoints to say that the sequence is wrong defined because it never ends. It is a valid philosophic aptitude, and in this way you deny the notion of convergence of a series, independently of the number the series is converging to. In Connes's case, series are converging to a very well known finite number, namely zero. You told me in a previous message that you were happy with the notion of convergence, but now it seems you confuse this notion with the one of infinitesimal and then you attack it. If you do not admit convergence, we can not proceed anymore in this mathematical way, and I am not sure if there is any other.

What Connes is telling there is that for instance the sequence

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,... (convergent to zero)

defines an infinitesimal of different size (order) that the sequence, for instance,

1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49,... (convergent to zero).

The only concepts involved are series and convergence of series.

Additionaly, Connes assumes that it is possible to extract a ordered series out of the eigenvalues of an operator. As I told, this could be resting on Zorn's Lemma if the series is to be extracted via an orthonormal basis, but I am not sure of the degree of generality required.

But I can not see any circularity in the arguments. I say more: there is no circularity in the argument. The infinitesimals defined by Connes are not used in the definition of convergence as you seem to believe in your uncareful reading. They are used against (in duality with?) Dixmier trace to build a theory of differentials and integration.

If you do not believe convergence, eg if you do not admit that classical mathematics can stablish that 1+1/2+1/4+1/8+1/16+1/32+1/64 --> 2 or that the decimal numbers 1.999999999999... and 2.0000000000000... are the same real number, then of course you can not admit Connes's aproach. I entered in this discussion mistaken by your previous affirmation about not having problems with convergence.

Dejame añadir, por estar seguro de que se me sigue, que las dos series que pongo en este mensaje son ejemplos exactos (no metaforas ni nada por el estilo) del tipo de series que Connes esta describiendo en los parrafos que he citado. Las series no usan ningun infinitesimal ni convergen a un infinitesimal (de hecho convergen a cero). Las series *son* los infinitesimales.

I think that you are not fixing the point. I think that you are failing to understand infinitesimal concept. Perhaps, the error is that i am explaining bad to you. I will atempt again.

The sequence goes to zero but cannot be zero for infinitesimal calculus. It cannot be zero then but cannot be any real number different from zero because if goes to 0.000005, exists any 0.000000005. Then would I put directly zero? No, it cannot be zero. The only explanation posible is a number between zero and any other small real number. The only possible number is an infinitesimal.

As perfectly stated by Connes the size of the infinitesimal (first order) is of order

(1/n)

but what is the value of n?

if n is any great real number (e.g. 10¹⁰⁰⁰) then (1/n) is NOT an infinitesimal, because an infinitesimal is more small that ANY real number and

10^-1000 is not more small than 10^-1001

There is not real number possible for n. Then one could work with n = infinite but then one obtains that infinitesimal would be

(1/n) = 0

**************

Note: Of course the infinitesimal is operationally defined via operators in Connes approach but the real infinitesimal is a number, it is not an operator.

When one wrote dx = v dt. dt is not an opperator. The situation is similar to QM, one defines the momentum operator but one is finally interested in its eigenvalue for comparison with experimental values.

**************

but an infinitesimal is by definition NON zero.

Therefore Connes writes the undetermined n ---> infinite.

Robinson calculus attacks directly that undetermination stating that n is a hiperreal number with inverse (1/n) = infinitesimal. n is not infinite but is more great that any real number.

Once explained this, you can see that Connes is defining the infinitesimal using the definition of infinitesimal in n ---> infinite. It is a circular definition. therefore of no interest for the understanding of infinitesimal. Robinson addresses this directly via his nonstandard analysis.

I can understand how the real series

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,... (convergent to zero)

converges to zero. But convergence is defined via the use of real numbers and the concept of infinite. There exits not infinitesimal defined in the real series of above. But if you want define the infinitesimal dp, which is not a real number, then you may find a number which is not zero, but is more small that any real member 1/n of above series

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,... 1/n,... dp,... 0.

The notation bold means that element does not belog to series because series is real one but dp is not a real.

Connes cannot do n = infinite but cannot do n = greatest real number then he use the standard but imprecise definition n ---> infinite. Exactly Connes would use n = (1/dp)

This is reason which both infinitesimals and their inverses are hiperreal numbers in nonstandard analisys. The topology is

infinite > (1/dp) > Real number - {0} > dp > 0

Note that n ---> infinite that Connes uses is included in the non real part of above topology.

The circularity is that Connes is REALLY using

n = (1/dp)

for defining dp.

Of course, Connes ignores this imprecission by stating the old concept of "n --> infinite". But n is not any real number of above series and (1/n) is not any memeber of series. n is not defined in Connes approach the statement

"n --> infinite" only says that n is "close" to infinite but n =/= Real and n =/= infinite. Since size of infinitesimal (and infinitesimal itself) is defined via n, standard analysis cannot correctly understand the concept of infinitesimal. The same error arises in the standard theory of limits.


The decimals numbers 1.999999999999... and 2.0000000000000... are the same real number, if the number of digits is infinite. But then the diference between both is zero and infinitesimals are not zero.

There are infinitesimals surrounding 2 but are not 1.9999999999...99 or 2.00000000000000...01 which are both real numbers.

The infinitesimals are (2 + dp) and (2 - dp).

Moreover questions like what is the exponential of... what are hard to reply on nonstandard calculus

are solved in my epsilon calculus.

 

[arivero]

I think that you are not fixing the point. I think that you are failing to understand infinitesimal concept. Perhaps, the error is that i am explaining bad to you. I will atempt again.

I can tell you that I understand the objections you formulate about the use of the infinite and the infinitesimal in mathematical practice. The only thing I am disagreeing is about if such objections can be applied directly in the parragraphs of Connes's work we are discussing about.

My impression is that the generality of such objections carries you to think there should be present also in this work, and then you are reading into the text instead of from the text. Anyway I think we have both exposed our interpretations of the text and any third reader could decide by herself by reading them. From my part, any further prolongation of the thread whould be simple repetition or, at most, rewording.

 

[Juan R.]

I can tell you that I understand the objections you formulate about the use of the infinite and the infinitesimal in mathematical practice. The only thing I am disagreeing is about if such objections can be applied directly in the parragraphs of Connes's work we are discussing about.

My impression is that the generality of such objections carries you to think there should be present also in this work, and then you are reading into the text instead of from the text. Anyway I think we have both exposed our interpretations of the text and any third reader could decide by herself by reading them. From my part, any further prolongation of the thread whould be simple repetition or, at most, rewording.

In short, what is n in Connes approach?

It cannot be infinite because then he would use the notation n = infinite but uses the standard notation "--->" what means in words "infinitesimally close to". In fact if Connes had used n = infinite then (1/n) = 0 but the size of an infinitesimal may be different of zero.

but n cannot be any real number, because the size of an infinitesimal is not (1/n) for any n real.

If cannot be infinite and cannot be real what is then n?

a) hiperreal number or

b) epsilon number if my approach is correct.

And YES this thread is very large, i am exhaust, but discussion with you was very interesting.

 

[janusz]

What about rate of expansion in empty intergalactic space ?-,to my knowledge it is faster .In present of matter slower,in present of movement again slower.Is it prove of interaction between gravity and space expansion? Why always on the charts -line -symbolise time,why is not flat membrane penetrating space constantly in every direction,point of spacetime curled up in within matter expanding ,spreading to the new points and giving out power natural colapsing matter to hold the structure.No time travel,no speed faster then light no wormholes.Nothing can penetrate the boundry of expansion even black hole slowly evaporate back.Do we really need this army of gravitons?

 

[OnTheCuttingEdge2005]

I believe from my own stand point that gravity is a product of the Conservation of energy theory and is also related to displacement theories hence combining these is what I see to be Gravity, I do not see Gravitons as an expotential of gravity, I have serious doubt about the existence of Gravitons, If I am to be flamed for not believing in gravitons, then so be it!

Displacement Theories: http://www.math.sintef.no/geoscale/slides/uib04.pdf

Laws of Conservation: http://en.wikipedia.org/wiki/Conservation_of_energy

Gerald L. Blakley

 

[flotsam]

How can Gravity have particles. Gravity is simply the curvature of spacetime. When a body is attracted to a larger body, its just following contours and curves formed. Is space made up of gravitons or When a large object curves it, automatically gravitons spread in the area.

I have asked myself this same question. Why even speculate about the existence of the graviton, never mind search for it. I recently read 'The fabric of the cosmos' and was dissapointed that when this apparent paradox came up, Brian Greene neglected to (for a layman like myself) explain this.

GR predicts gravitational waves but I would have thought that these waves in themselves are just the warping of spacetime and not a force.

 

[-Job-]

And gravitons being particles, would they have gravity?

 

[Pippo]

Apologize, I have deleted my messages,hope I will get familiar with this posting.

 

[janusz]

And gravitons being particles, would they have gravity?

Profesional phisics found ether and now they are feeding us with the particle for every force of nature.S. Hawking talks about final theory - is it something similar happened in XIX century?
We are lucky that is still something like freedom of speculation ,very healthy.

 

[janusz]

I have asked myself this same question. Why even speculate about the existence of the graviton, never mind search for it. I recently read 'The fabric of the cosmos' and was dissapointed that when this apparent paradox came up, Brian Greene neglected to (for a layman like myself) explain this.
GR predicts gravitational waves but I would have thought that these waves in themselves are just the warping of spacetime and not a force.

I think that the gravity is the obstacle for unification of the forces,thats why everybody try to find brackets of grasping it .If it is like you say we are looking for transmission of curvature of the spacetime , I wonder if it is possible to find it while spacetime itself is expanding . For example in thought experiment enlarge your body enough to put your hand in between the molecules ,when you move ---the spacetieme within you does not stay the same----it is not your property,your body occupay different part of the spacetime in every split of the second and this spacetime swelling.What about if we add some extraordinary speed and time shrank,are we occupaing more of the space in that split of second ,what happened to the gravity of the system of your body ,it has to has some of the curvature ,wraping space ability?