How can Gravity have particles.

[Caesar_Rahil]

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.

 

[Berislav]

How can Gravity have particles. Gravity is simply the curvature of spacetime.

General Relativity predicts the existence of gravitational waves. Thinking by analogy one can conclude that waves must be in fact a bunch of mediator particles. Hence the graviton.

 

[hellfire]

However, there must be a bit more in gravitation than this conceptual step from a perturbation on a fixed spacetime to the graviton. The graviton is a particle and particles are defined according to their transformation properties on flat spacetime. I do not think the graviton will be a good explanation of the gravitational phenomenon in a generic spacetime without symmetries... but don’t take my opinion too seriously.

 

[Juan R.]

However, there must be a bit more in gravitation than this conceptual step from a perturbation on a fixed spacetime to the graviton. The graviton is a particle and particles are defined according to their transformation properties on flat spacetime. I do not think the graviton will be a good explanation of the gravitational phenomenon in a generic spacetime without symmetries... but don’t take my opinion too seriously.

Absolutely correct!

Graviton arises only in the linear regime perturbation expansion of GR. Of course, one can still think of a nonlinear field mediated by gravitons in the full perturbative regime. But their definition is not rigorous when one abandons flat spacetime.

The reason of that a graviton-like behavior is observed in string theory is precisely that causality is defined on a flat regime violating GR.

Note: Graviton is speculative.

 

[marcus]

How can Gravity have particles? Gravity is simply the curvature of spacetime...

Yes, that is a good thing to have noticed, Caesar. I have the impression in the case of many physicists, the graviton is a SOMETIMES USEFUL (but also sometimes not useful) mathematical device, that applies in certain gravity situations.

for example if the gravitational field is weak, so that the spacetime is nearly flat or in some other way very regular, then one can imagine doing an APPROXIMATE analysis by "perturbation" method. Then the "graviton" is defined and makes sense as a quantum of the field and it would be mathematically very useful

But on the other hand, there would, I imagine, be other situations where the geometry is not nearly flat or simple in some other easily described way, but where the geometry is highly curved or irregular and where the "perturbation" method of approximation does not apply. In that case the graviton would not be useful idea. One would have to deal directly with the geometry, and one would not pretend that the gravitons exist, in that situation.

So I suppose for many quantum gravity physicists, the graviton does not exist in any absolute sense, but only has a kind of conditional applicability---it works as an idea in some situations but not in others.

Another point to make is what ENERGY do you imagine typical for the gravitons of the world. If it is a binary pulsar with orbital period of 6 hours then it is a very low frequency wave! General Rel describes the wave, that part is OK. but the wave is very low frequency compared with light and radio waves. But the field quantum energy (the "graviton" energy) is proportional to the frequency, so the energy of an individual quantum (that one might wish to observe by some means analogous to the Photoelectric Effect) would be miniscule. It might be instructive to estimate the typical energy of the gravitons which one supposes are flying around us, if one wants to think of it that way.

human theory (like choice of language) is somewhat an area of freedom and we should be free to model reality as we like, but then one needs to check out the consequences and the range of applicability where it works

OOPS I see now I have duplicated some things already said by other people

 

[Ratzinger]

but do/would gravitons quantize spacetime? (whatever that means, just read it once on PF)

 

[marcus]

but do/would gravitons quantize spacetime? (whatever that means, just read it once on PF)

I think "gravitons quantize spacetime" is nonsense. It does not mean anything and is misleading. I did not hear anyone at PF say this.
Probably if one wants to talk about the fundamental nature of spacetime geometry, then one should not bother to talk about "gravitons"

they are more of a mathematical fiction or a useful device in some situations. they belongs more to a perturbative approach to gravity

and not to a quantum theory of spacetime geometry.

 

[Pippo]

My idea is that gravitation is a result for the masses to save energy in creating the space.
Giuseppe

 

[Juan R.]

Another point to make is what ENERGY do you imagine typical for the gravitons of the world. If it is a binary pulsar with orbital period of 6 hours then it is a very low frequency wave! General Rel describes the wave, that part is OK. but the wave is very low frequency compared with light and radio waves. But the field quantum energy (the "graviton" energy) is proportional to the frequency, so the energy of an individual quantum (that one might wish to observe by some means analogous to the Photoelectric Effect) would be miniscule. It might be instructive to estimate the typical energy of the gravitons which one supposes are flying around us, if one wants to think of it that way.

Well, some decades ago almost all quantum gravity researches believed that graviton was quantum gravity.

Precisely, Dyson did a criticism about string theory where emphasize the fact that gravitational waves are not observed and, therefore, graviton is a pure mathematical device with no direct physical insight even in the perturbative regime.

He speculates that gravitons really does not exist. He claim that even if really gravitons exists we cannot detect it, doing their existence outside of physics (belonging just to metaphysics).

I agree, it may be a good advice to do physics for observed stuff only

 

[Juan R.]

but do/would gravitons quantize spacetime? (whatever that means, just read it once on PF)

No!

At the best, gravitons would "quantize" the curvature of spacetime, that is, gravity.

The best example is superstring theory. It does an attempt to "quantize" curvature of spacetime using a graviton that arises of a string vibration mode. However spacetime in string theory is not quantized, in fact is described by a Calabi-Yau, it which is classical manifold. This is incorrect.

However, other approches to QG quantize spacetime, e.g. loop quantum gravity,obtains quantums of area and volume. This is correct.

 

[kneemo]

However, other approches to QG quantize spacetime, e.g. loop quantum gravity,obtains quantums of area and volume. This is correct.

Quantizing geometry is an ongoing mathematical program that is not yet complete. To place this in context, imagine how difficult it would be for Einstein to formulate General Relativity without Riemannian geometry. A big applause to the brave LQG and CDT researchers thus far!




Sir Michael Atiyah (Abel Prize 2004):

"Alain Connes' program is very natural - if you want to combine geometry with quantum mechanics, then you really want to quantize geometry, and that is what non-commutative geometry means."

 

[Juan R.]

Quantizing geometry is an ongoing mathematical program that is not yet complete. To place this in context, imagine how difficult it would be for Einstein to formulate General Relativity without Riemannian geometry. A big applause to the brave LQG and CDT researchers thus far!


Sir Michael Atiyah (Abel Prize 2004):

"Alain Connes' program is very natural - if you want to combine geometry with quantum mechanics, then you really want to quantize geometry, and that is what non-commutative geometry means."

However, at my best current knowledge, nobody has developed a consistent quantization of geometry from NC geometry. In fact, nobody has shown that non commutative space or spacetime is related to the Planck.

Nobody has found the correct theta, etc.

Moreover, NC geometry just introduces non-commutativity between pairs of observables and says that a single observable would be classical. I cannot agree with Atiyah (even if he is one of best mathematicians of 20th!). It is not clear for me that

non-commutativity = quantize

Take for example the quantum [p_op, x_op] =/= 0

However the spectra of operator x_op is continuum. x has been not quantized.

Non commutatividad of (x, p) space expressed in a star product f(x)*f(p), for arbitrary functions, does not imply that x was quantized. In fact, the spectral decomposition of x_op is classical one.

LQG really "quantizes" spacetime (x, t) and obtains the quantum of area and volume. From my own methods, i also obtain quantum of areas and volume. In fact, from a simple argument, it is easily shown that from a pure differential manifold both BH cannot radiate and the law of increase of horizons is NOT satisfied due to collapse of differential geometry.

ST "predicts" entropy of BH from a CY manifold but fail to explain the evolution of BH because the topology of S functional is NOT studied in ST.

This is another proof of that ST is a waste of time.

 

[arivero]

I cannot agree with Atiyah (even if he is one of best mathematicians of 20th!). It is not clear for me that[/B]

non-commutativity = quantize

I am not sure that Atiyah is claiming that this; non-commutativity is surely more general.

There was an intuition that Bohr-Sommerfeld quantisation was a kind of index theorem, sort of a Chern class. And there was the point that non-commutativity generates very straighforwardly this kind of index theorems.

Then there happened the observation that the fields in the standard model can be rearranged into non-commutative fields fullfilling the axioms of a non commutative geometry.

And very recently it was shown that Moyal plane also fulfils such axioms:
http://arxiv.org/abs/hep-th/0307241

Take for example the quantum [p_op, x_op] =/= 0

However the spectra of operator x_op is continuum. x has been not quantized.

Non commutativity of (x, p) space expressed in a star product f(x)*f(p), for arbitrary functions, does not imply that x was quantized. In fact, the spectral decomposition of x_op is classical one.

LQG really "quantizes" spacetime (x, t) and obtains the quantum of area and volume.

Exactly: x is not quantised in quantum gravity. There are not such a thing as a discretised coordinate, just discretised areas and volumes.

This is not so strange to me nor to any geometer; the line has not intrinsic properties, the surface has. To a geometer, there is nothing to quantise in a line.

Also, note that Planck length is just a convention modulo order of magnitude, because after all G is an area. One could define Planck Length as the radius generating a circle for this area, or the diameter, or the diagonal of the square having such area, or the side of the square, etc.

 

[Juan R.]

I am not sure that Atiyah is claiming that this; non-commutativity is surely more general.

Yes, i would have wrote

non-commutativity => quantize

However non-commutativity is not more general. For instance, in a non-commutative sense, a single coordinate is non quantum for a physicist but quantum for a geometer. See arXiv:gr-qc/9803024 pag 12.

A source of confusion about noncommutative geometry is the use of the expression “quantum”. In the mathematical parlance, one uses the expression “quantization” anytime one replaces a commutative structure with a noncommutative one, whether or not the non-commutativity has anything to do with quantum mechanics.

I already explained that (x, p) space is noncommutative when x continues being a classical magnitude. From my point of view, quantization of spacetime is more general than noncommutativity of coordinates.

Exactly: x is not quantised in quantum gravity. There are not such a thing as a discretised coordinate, just discretised areas and volumes.

This is not true. The reason of emphasis on areas and volumes is that it is more simple to construct the operator from the mathematical side, but at my best current knowledge LQG predicts that length is also quantized.

 

[Sam Owen]

I imagine gravity to be the surface tension of light in connected vacuum filled bubbles making quantum foam

that implies light has mass, the bubble at Planck level resmbles a geodesic sphere and at universal level resembles the universe we see

remembering bubbles always connect by triangles and photons are the fundamental particles

like soccer balls connected by sharing the same patch which then opens up to another part of the universe or another universe and at Planck level connects to another sphere of light

everything is made of light that blinks in and out of 2 sets of 3 dimensions oppositely charged at superluminal speed

and we can't tell the difference merely registering the effect through the illusion of perceived observation

 

[arivero]

This is not true. The reason of emphasis on areas and volumes is that it is more simple to construct the operator from the mathematical side, but at my best current knowledge LQG predicts that length is also quantized.

It could be, but I am not aware of the existence of an operator for length taking only discrete values and I would be grateful if someone surprises me by referencing a paper doing so. I have heard only of area operators.

Furthermore, the whole issue of spin foams is about intersecting surfaces so that the area becomes a multiple of the number of intersecting lines. So it is a conceptual point, no a technical one.

 

[arivero]

However non-commutativity is not more general. For instance, in a non-commutative sense, a single coordinate is non quantum for a physicist but quantum for a geometer. See arXiv:gr-qc/9803024 pag 12.

You have misunderstood the point in this paper. It simply tells that the quantum space of Connes-Lott model is not a model of quantum field theory, but of classical field theory. By "non quantum for a physicist" it means "not quantum mechanical". Connes-Lott are a family of models over midly noncommutative spaces (spectral triples) that can be reinterpreted as classical fields over commutative ones. It is not speaking about how "single coordinates" could be quantised.

Now, Moyal plane (* product) can be also recast as a spectral triple, a recent result that Rovelli can not quote there. Thus spectral triples include also quantum mechanics.

 

[Juan R.]

It could be, but I am not aware of the existence of an operator for length taking only discrete values and I would be grateful if someone surprises me by referencing a paper doing so. I have heard only of area operators.

Furthermore, the whole issue of spin foams is about intersecting surfaces so that the area becomes a multiple of the number of intersecting lines. So it is a conceptual point, no a technical one.

as said, it is! See arXiv:hep-th/0408048

The area, volume and length operators have discrete, finite spectra, valued in terms of the Planck length. There is hence a smallest possible volume, a smallest possible area, and a smallest possible length, each of Planck scale. The spectra have been computed in closed form.

 

[Juan R.]

However non-commutativity is not more general. For instance, in a non-commutative sense, a single coordinate is non quantum for a physicist but quantum for a geometer. See arXiv:gr-qc/9803024 pag 12.

You have misunderstood the point in this paper. It simply tells that the quantum space of Connes-Lott model is not a model of quantum field theory, but of classical field theory. By "non quantum for a physicist" it means "not quantum mechanical". Connes-Lott are a family of models over midly noncommutative spaces (spectral triples) that can be reinterpreted as classical fields over commutative ones. It is not speaking about how "single coordinates" could be quantised.

Now, Moyal plane (* product) can be also recast as a spectral triple, a recent result that Rovelli can not quote there. Thus spectral triples include also quantum mechanics.

I explained very bad.

My post #14 would say "See arXiv:gr-qc/9803024 pag 12 for the next quote".

Previous phrase that you cite above is my own. It is not related to the preprint cited. I will explain again my point.

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).

If you work with * products, non-commutativity looks like quantum mechanics, if your correctly link non-commutative parameters theta with Plack h. Which is not needed in mathematics, that is theta can be perfectly classical (h -> 0, Theta =/= 0). Physicists call quantum only when Planck h is non zero.

But a single function is not modified in NC geometry. For example asume that theta = theta(h), the Moyal bracket verifies

f(x_NC)*1 - 1*f(x_NC) = 0

and f(x_NC)*1 = f(x_NC)·1, and the result is equivalent to that of commutative space (classical for a mathematician). In fact, for above function f one can see that x_NC = x

This is why noncommutativity of the space is non equivalent to a pure quantization of x because f(x_op) =/= f(x) with direct quantization and x_op =/= x, doing it more general.

This is more easy to see remembering that quantization means that observable cannot take any value (only a discrete set of values) whereas Moyal "quantization" or NC geometry means that two complementary observables cannot take definite values at once.

As already explained non commutativity of phase space (x, p) does not mean that x was a quantum magnitude and in fact is not because spectrum of possible values for x is perfectly classical one.

Moreover, there are fundamental difficulties in NC geometry that have been not adressed still. For example, the definition of differentials is not correct for me.

I expect (Thanks Marcus!) you understand better now my point.

 

[arivero]

as said, it is! See arXiv:hep-th/0408048

I am surprised!

Well, the quote in this paper is mostly a redirection to Thiemann's gr-qc/9606092 "A length operator for canonical quantum gravity". Check around pages 15 and 16 there to see that even if correct, it is not a very intuitive "quantum of length" because it involves both length and angular momentum or spin. Such dependence is not to be very wellcome and it raises doubts (Thiemann himself at the end of the paper) about the classical limit.

Is it the only paper building such quantum? In page 4, Thiemann discuss why, in his opinion, such operator had been not build until then. And the papers quoting Thiemann are mostly general reviews and they do not seem to make any use of the operator.

 

[arivero]

I explained very bad.

My post #14 would say "See arXiv:gr-qc/9803024 pag 12 for the next quote".

Previous phrase that you cite above is my own. It is not related to the preprint cited. I will explain again my point.

Er, then why did you cite the paper, if it is unrelated to the context?

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.

Question, do you see quantisation as a kind of discretization? For instance, should you be happy with a discrete spectrum say 1/n, or as the discrete part of the energy spectrum of Coulomb potentials? Or besides a discrete spectrum do you want a minimum quanta?

LQG quantizes spacetime including single coordinates (lengths).

Well, OK, Thiemann does.

This is more easy to see remembering that quantization means that observable cannot take any value (only a discrete set of values) whereas Moyal "quantization" or NC geometry means that two complementary observables cannot take definite values at once.

As already explained non commutativity of phase space (x, p) does not mean that x was a quantum magnitude and in fact is not because spectrum of possible values for x is perfectly classical one.

Note that it is a very bold position to say that the position or momentum operator in quantum mechanics are not quantum quantities. It seems that your point is that the spectum is not discrete, but it creates a bit of confusion.

On other hand, it is interesting to look into quantum mechanics for the interplay between quantisation and complementary observables. Because two complementary observables will forcefully have the units of the quantised one (Famous paradox here, is to look for the complementary observable of a spin or an angular momentum operator).

Moreover, there are fundamental difficulties in NC geometry that have been not adressed still. For example, the definition of differentials is not correct for me.

Hmm from the above I suspect you assimilate NCG to the Moyal structure, this is only an aspect of NCG, very popular due to its use in string theory. In Connes theory, one can define differentials in an abstract way via universal algebra, or in a very concrete way via Fredholm Modules (commutator [F,a], when you are only interested on differentiation) or Spectral Triples (commutator [D,a], when you also want a metric structure). Besides, there are a different way to quantum differentials lead my Majid, in his approach to quantum groups.

 

[Juan R.]

I am surprised!

Well, the quote in this paper is mostly a redirection to Thiemann's gr-qc/9606092 "A length operator for canonical quantum gravity". Check around pages 15 and 16 there to see that even if correct, it is not a very intuitive "quantum of length" because it involves both length and angular momentum or spin. Such dependence is not to be very wellcome and it raises doubts (Thiemann himself at the end of the paper) about the classical limit.

Is it the only paper building such quantum? In page 4, Thiemann discuss why, in his opinion, such operator had been not build until then. And the papers quoting Thiemann are mostly general reviews and they do not seem to make any use of the operator.

I think that already said that there are mathematical difficulties

The reason of emphasis on areas and volumes is that it is more simple to construct the operator from the mathematical side, but at my best current knowledge LQG predicts that length is also quantized.

but there is no reason why the length operator cannot exist. There are difficulties for definition of QG but, a priory, it may exist.

 

[Juan R.]

Er, then why did you cite the paper, if it is unrelated to the context?

Unrelated to previous phrase related to next. Already said that was an error. Instead of "See arXiv:gr-qc/9803024 pag 12" would say "See arXiv:gr-qc/9803024 pag 12 for the next quote".

Question, do you see quantisation as a kind of discretization? For instance, should you be happy with a discrete spectrum say 1/n, or as the discrete part of the energy spectrum of Coulomb potentials? Or besides a discrete spectrum do you want a minimum quanta?

Are not equivalent, but a quantization of spacetime would introduce quantums of space and time via some kind of discrete eigenvalue problem. In NCG a length is a continuum magnitude. There are not chronons on NCG.

Note that it is a very bold position to say that the position or momentum operator in quantum mechanics are not quantum quantities. It seems that your point is that the spectum is not discrete, but it creates a bit of confusion.

Of course, the position operator in quantum mechanics is a quantum quantity. But x_op = x, and spectrum of x is continuum. Compare that with usual p_op =/= p.

In a quantization of spacetime x_op =/= x, and one obtains quantums of position (area, and volume) as one obtains today quantums of energy and momentum in standard QM. Precisely the failure of QFT to describe quantum gravity is that spacetime is classical (no quantum) manifold.

Hmm from the above I suspect you assimilate NCG to the Moyal structure, this is only an aspect of NCG, very popular due to its use in string theory. In Connes theory, one can define differentials in an abstract way via universal algebra, or in a very concrete way via Fredholm Modules (commutator [F,a], when you are only interested on differentiation) or Spectral Triples (commutator [D,a], when you also want a metric structure). Besides, there are a different way to quantum differentials lead my Majid, in his approach to quantum groups.

The Moyal structure was already very popular before string theory. It is standard in statistical mechanics of semiclassical systems. I was talking of Connes algebraic procedure for defining differentials via condition n --> infinite that he uses on compact operators. It is difficult for me understand infinitesimals on that way.

 

[arivero]

I was talking of Connes algebraic procedure for defining differentials via condition n --> infinite that he uses on compact operators. It is difficult for me understand infinitesimals on that way.

Ah, ok, yep it is difficult . The goal in this procedure is to exhibit explicit infinitesimals, to have an infinitesimal object you can calculate with, and an compact operator works nicely especially if you compare it to other approaches (e.g., ultrafilters as used to explicit Robinson calculus). Combined with the trace, the compact operator behaves perfectly well as an infinitesimal, performing the integral only when it is order one.

 

[Spin_Network]

I am not sure that Atiyah is claiming that this; non-commutativity is surely more general.

There was an intuition that Bohr-Sommerfeld quantisation was a kind of index theorem, sort of a Chern class. And there was the point that non-commutativity generates very straighforwardly this kind of index theorems.

Then there happened the observation that the fields in the standard model can be rearranged into non-commutative fields fullfilling the axioms of a non commutative geometry.

And very recently it was shown that Moyal plane also fulfils such axioms:
http://arxiv.org/abs/hep-th/0307241




Exactly: x is not quantised in quantum gravity. There are not such a thing as a discretised coordinate, just discretised areas and volumes.

This is not so strange to me nor to any geometer; the line has not intrinsic properties, the surface has. To a geometer, there is nothing to quantise in a line.

Also, note that Planck length is just a convention modulo order of magnitude, because after all G is an area. One could define Planck Length as the radius generating a circle for this area, or the diameter, or the diagonal of the square having such area, or the side of the square, etc.

The nearest thing is this:http://arxiv.org/abs/gr-qc?0407022

?

 

[Juan R.]

Ah, ok, yep it is difficult . The goal in this procedure is to exhibit explicit infinitesimals, to have an infinitesimal object you can calculate with, and an compact operator works nicely especially if you compare it to other approaches (e.g., ultrafilters as used to explicit Robinson calculus). Combined with the trace, the compact operator behaves perfectly well as an infinitesimal, performing the integral only when it is order one.

However, is not the infinitesimal of the spectra decomposition defined via the limit on the compact operator? How does Connes define the limit n --> infinite? I think that his logical approach is basically as follow:

1) define limit n --> infinite via classical theory of limits.

2) Define theory of operators

3) apply limit n --> infinite to operator for obtaining the definition of infinitesimal.

My problem is already with (1). Because usual theory of limits is not rigorous and already use (in implicit form) the definition of infinitesimal. Precisely the classical theory of limits was introduced in the 19th for avoiding the use of infinitesimals but they continue therein.

I can agree with you that Connes approach is more useful in many ways that non standard calculus, but i work via a different method. I use a new calculus that i call epsilon-calculus and i am developing.

I defined for my own use in physical questions: epsilon structures, (1/R) duality, dissipative Hamiltonian equations, topology of differential geometry in thermal states, Poincaré resonances on large quantum systems (LPS), etc.

However, it is also useful for mathematical questions. There exists some link with nonstandard analysis but it appears more applicable. For example, Connes claims that from Robertson calculus one can reply simple questions about infinitesimals but more hard questions how

what is the exponential of –1/dp(x)?

are not solved by Robertson or others mathematicians. From epsilon calculus, i obtain the next reply in two steps

exp[–1/dp(x)] = 0 with dp(x) =/= 0

 

[arivero]

However, is not the infinitesimal of the spectra decomposition defined via the limit on the compact operator? How does Connes define the limit n --> infinite?

Implicitly, via the available theory of representation of C*-Algebras in Hilbert spaces. Thus it inherites the strongness and weakness of Hilbert space theory.

However, it is also useful for mathematical questions. There exists some link with nonstandard analysis but it appears more applicable. For example, Connes claims that from Robertson calculus one can reply simple questions about infinitesimals but more hard questions how

what is the exponential of –1/dp(x)?

are not solved by Robertson or others mathematicians. From epsilon calculus, i obtain the next reply in two steps

exp[–1/dp(x)] = 0 with dp(x) =/= 0

In Connes, dp is an operator A in Hilbert space, so Exp(-A^-1) is well defined. That was the whole point of the comparision against Robinson.

 

[Juan R.]

Implicitly, via the available theory of representation of C*-Algebras in Hilbert spaces. Thus it inherites the strongness and weakness of Hilbert space theory.

Then what is n? Is n a real or complex number? is infinite? That is not sufficiently rigorous for me! This is the reason of theory of limits used by Connes is not sufficient and i developed epsilon calculus.

In Connes, dp is an operator A in Hilbert space, so Exp(-A^-1) is well defined. That was the whole point of the comparision against Robinson.

Great! I simply said that from epsilon calculus i can offer reply that mathematicians as Robertson cannot.

 

[arivero]

But you get zero directly; are you sure it is the right result?. This example is not chosen randomly, isn't it? If dp is -say a la Robinson- an infinitesimal real, then 1/dp is an infinite real, and exp(-1/dp) is the minus exponential of an infinite quantity. Should it be zero, or should it be an infinitesimal of some increased order? What about log(exp(-1/dp)) in your formulation? And about 1/log(exp(-1/dp))?

(Juan, estas tu equivocado en el nombre o lo estoy yo... ¿Es Robertson o Robinson el que invento el Analisis No Estandar? )

 

[arivero]

Then what is n? Is n a real or complex number? is infinite? That is not sufficiently rigorous for me! This is the reason of theory of limits used by Connes is not sufficient and i developed epsilon calculus.

Well, that means that Hilbert Spaces are not sufficiently rigorous for you. Not blame on this. The existence of a basis in a generic, possibly infinite dimensional, Hilbert Space depends on the Axiom of Choice.