Does the Inverse Square Law Justify Compactification to 3+1 Dimensions?

[arivero]

I mean, beyond the physical argument... has people (scientists) worried about to justify why the configuration space of Nature should compactify to 3+1?

In a previous thread, we found (we=marcus+me+orion+rest of readers if any) that gravitational coupling constant simplifies out of some equations only if the force law is inverse square.

This rings a bell: inverse square law is, arguably, consistent only with 3D space. Thus any "dimension-independent proof" of inverse square law should also justify us to compactify into 3D space. So, second question... any papers on this?

 

[arivero]

Synge

The matematician JL Singe published in 1935 a PhysRev paper,
J.L. Synge, Angular Momentum, Mass Center and the Inverse Square Law in Special Relativity, Physical Review 47, p 760
There, he raised the idea of linking the potential energy V(x), to the total energy of photons exchanged. He got some minor results, only for repulsive interactions and with a lot of assuptions, but pointing to a inverse-square force law independent of space dimensionality.
Also, he suggested (I can not see if it relates directly to his equations) a rule of thumb: "As the distance increases, the impulses arrive less frequently at A: this makes the momentum received by unit of time vary as 1/r. Secondly, the impulses become weaker as they impart energy to A: this produces another factor 1/r, and the combination of the two gives the inverse square law"If one thinks a little bit, a similar insight could be derived from quantum mechanics plus relativity:

On one side, we can request that the exchanged particle has a wavelength similar to the separation between the interacting particles. This gives a factor 1/r to the exchanged momentum.
On other side, we can require the transference time to be of the order of "De Broglie time" of the carrier, or similarly just t=r/c. This give us the second 1/r factor, as force is dp/dt.

 

[ranyart]

It all boils down to this:https://www.physicsforums.com/showthread.php?s=&postid=150690#post150690

One can clearly see why Einstein sought out Geometry, what he found in Geometry became very relevant to his understanding of GR.

 

[arivero]

Ranyart, the low-dimensional work you suggest is the kind of thing I was alluding with Galois Theory, middle way in other thread. Only that the real thing seems, to me, a very complicated issue with a lot of group theory. Besides lacking a clear formulation of the problem.

 

[ranyart]

Originally posted by arivero
Ranyart, the low-dimensional work you suggest is the kind of thing I was alluding with Galois Theory, middle way in other thread. Only that the real thing seems, to me, a very complicated issue with a lot of group theory. Besides lacking a clear formulation of the problem.

Arivero I do believe that Nash : http://en.wikipedia.org/wiki/Nash_embedding_theorem

produced some kind of technique leading to a thereom of foldings ie:http://en.wikipedia.org/wiki/Convolution

It may be that the smallest geometric space is the Un-folding(non-curveture) of a smallest amount of matter that can exist with mass in 3-D environment (mass cannot be observed/exist in less than 3-Dimensions), the obvious candidate being Quarks bounded as Protons.

The distribution of Quarks throughout the Universe during Supernova's are not observed until they connect into a 3-D enviroment. The transportation for Quarks is that they are converted 'before' flight takes place, they are Dimensionally reduced within their 'local' environment into EM 'bits', thereby no Mass-Gain (SR) on their journey across Extra Galactic Space, until they encounter our Galaxy's Halo.

There is a similar theoretic explination for 'in-flight' Neutrinos, which have three possible modes of existence. One can see that Einsteins E=MC2 is a Conversion of 3-d >> 2-d mass/field equation.

Collisions are more probable in 3-dimensions, and are less likely in 2-dimensional fields, because of the lack of Mass, any 2-D Quanta that has mass would be producing a curveture of space, and would therefore be in 3-D space.

Mass and Movement do not fare well in Relativity Frames, the best way to commute throughout the Universe is to loose your restraining baggage ie, Mass! Leaving your mass in 3-D space ensure's that you can always arrive at the SR station before the train leaves, you would always arrive, 'in-time' !

 

[arivero]

Originally posted by ranyart
Arivero I do believe that Nash : http://en.wikipedia.org/wiki/Nash_embedding_theorem

produced some kind of technique leading to a thereom of foldings ie:http://en.wikipedia.org/wiki/Convolution

It may be that the smallest geometric space is the Un-folding(non-curveture) of a smallest amount of matter that can exist with mass in 3-D environment (mass cannot be observed/exist in less than 3-Dimensions), the obvious candidate being Quarks bounded as Protons.

Again, it could be a small hint here around in your words, but not easy to concrete. Fact is, the Nash embedding theorem does not apply to (1,-1,-1,-1) metrics such as general relativity. In this case the bounds for an embedding are a lot higher, about ninety or one hundred degreees of freedom -I put concrete number in s.p.r. years ago-, or call it dimensions if you want.

 

[arivero]

Originally posted by ranyart
One can see that Einsteins E=MC2 is a Conversion of 3-d >> 2-d mass/field equation.

I am afraid I can not see it.

 

[ranyart]

Originally posted by arivero
I am afraid I can not see it.

Arivero, it is well know that when Bohm went to work with Einstein, there was a change of direction in Bohm:http://en.wikipedia.org/wiki/David_Bohm

The work in dealing with the 'Bohr Correspondence limit' Limit:http://en.wikipedia.org/wiki/Correspondence_principle

led Bohm to announce:The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit. For this reason, Bohm has argued that classical physics does not emerge from quantum physics in the same way that classical mechanics emerges as an approximation of special relativity at small velocities; rather, classical physics exists independently of quantum theory and cannot be derived from it.

Now I can wave you towards a number of factors that is allready contained in Einsteins work, especially 'In his Later Years'.

One can see that Einsteins E=MC2 is a Conversion of 3-d >> 2-d mass/field equation.

E=mc2 is dimensionally fixed, you cannot creat matter (3-D) from 'Mass/ENERGY that is allready 3-D' , what you do is alter an existing matter that remains in 3-D space.

All the creationary modes within E=MC2 can only happen when the convertion of a 'Target Matter' is reduced to an energy that is not 3-Dimensional. For instance a particle accelerator can 'create' excess matter particles because it is pulling energy from a 2-D field, matter-anti matter are results of combining existing 3-D energies 'target matter', colliding them together at high speeds, and fragmenting them into the smallest possible 3-D states.

It is when 3-D matter is finitely reduced, some of the energy is pushed into a 2-D field (allways the core of collision), and out 'pops' an excess amount of matter, eminating from a near-by 'field', which is why you cannot remove a Quark into isolation, as you try to take it from the 3-Dimensional state, it is constrained by the infinite amount of 2-Dimensional field energy.

Until the correspondence of Dimensionality is further understood with respect to GR and QM , then the Unification of GR with LQG and other theoretical studies, will always fall short.

The correspondence between Einstein and Bohr in the early days of Quantum theory, reminds me of current correspondence between String theorists and Loopists, it seems as if nothing in 100 years has changed!

If the 3-Dimensional world we are used to within our Galaxy, is the same 3-D space that lays between Galaxies, then Lorentz Transformation
would not be relevant, what makes it very relevant is the fact that traveling from one Galaxy to another, you would be encountering a 2-D space (EM-Vacuum) and this by its very existence would compel you to reduce your Dimensionality in order to proceed!

 

[arivero]

ranyart, instead of quoting authorities, simply note, as a counterexample, that special relativity can be defined in spaces beyond (and almost below, too) 3+1. Thus the equation E=mc2, as it is, does not depend practically of dimensionality.

Can you see it? If not, we will just agree to disagree.

You can incorporate dimensionality by putting more content... for instance asking special relativity to be a limit of general relativity.

[EDITED]: the content of the thread is about compactification, ie dimensionality in the sense of dimensions of space time. There is a near use of the word, "dimensional analisys", but the dimensional analysis of Lorentz (nor Einstein) equations per se does not give any clue of dimensionality of space.

 

[arivero]

synge-debroglie

Anyway, let's go back to this "DeBroglie" variant of Synge arguments, perhaps in this way it must be clear the kinf of relations I am asking for.

Consider a photon on-shell, and carrying energy and momenta between two sources at distance r.

From this separation r we can get a momentum $$\Delta p=\hbar/r$$ and then an energy $$E=pc=\hbar c /r$$ and then a corresponding time $$\Delta t=\hbar/E=r/c$$
Thus there exists a (dimension independent!) force
$$F={\Delta p \over \Delta t}= \hbar c {1 \over r^2}$$
and the propagator Green Function should be adjusted to fit on it. The only issue I see is that photons in virtual exchange are off shell, so we have a bound instead of an equality.

For massive mediators, the issue is trickier. To get an approximation to Yukawa force it seems one must use $$\Delta p=\frac1c \sqrt{E^2-m_0^2c^4}$$ and $$\Delta t= {r \over c}$$; I am not very easy about this second equation.

 

[ranyart]

Originally posted by arivero
Anyway, let's go back to this "DeBroglie" variant of Synge arguments, perhaps in this way it must be clear the kinf of relations I am asking for.

Consider a photon on-shell, and carrying energy and momenta between two sources at distance r.

From this separation r we can get a momentum $$\Delta p=\hbar/r$$ and then an energy $$E=pc=\hbar c /r$$ and then a corresponding time $$\Delta t=\hbar/E=r/c$$
Thus there exists a (dimension independent!) force
$$F={\Delta p \over \Delta t}= \hbar c {1 \over r^2}$$
and the propagator Green Function should be adjusted to fit on it. The only issue I see is that photons in virtual exchange are off shell, so we have a bound instead of an equality.

For massive mediators, the issue is trickier. To get an approximation to Yukawa force it seems one must use $$\Delta p=\frac1c \sqrt{E^2-m_0^2c^4}$$ and $$\Delta t= {r \over c}$$; I am not very easy about this second equation.

You may want to examine the equations on page three of this recent paper?

http://uk.arxiv.org/abs/astro-ph/0402554

Hope you find it of use.

 

[arivero]

dimensional analysis, ugh!

Thanks.

I have been thinking how all this boils down to dimensional analysis (Hi marcus ). At least in this way one can connect with the argument based on superficial divergences of renormalization theory.

Point is, that force has dimensions of [length]^-2. More precisely, one says that
$$[f]= [m][x]/[t]^2=[h]/[x][t]=[h][c]/[x]^2$$
where h and c are constant as usual, so the variable physical input must have a way to appear with dimension [length]^-2.

In absence of masses (remember $$[m]=[h][c]^{-1}[length]^{-1}$$) the only scale available is the separation between the interacting forces. So if we aim for a adimensional (scale-less) coupling constant, dimensional analisis imposes upon as a force following the inverse square law.

The same process sleeps at the end of the "degree of divergence" of a loop in perturbative QFT. It can be seen that the naive condition for renormalization is, simply, to compare the dimensionality of fields and coupling constants with the dimensionality of space.

In the following I'll put h=c=1 if only for training.
Lets try to add some masses but still insisting in scale-less coupling constant. First, a massive propagator "M" could be used just to reduce or to increase one degree of force. So we have the possibilities
$$f= K M/x$$, $$f=K 1/Mx^3$$.
But in such cases the limit $$M \to 0$$does not recover the previous forces (thus should we need to colapse one spatial dimension in the limit if we ask for consistency?). Still it is possible to use M just to cancel the distance in other way:
$$f={K \over x^2} (1-(M x)^p)^q$$
If q=1/2 then we have an interaction that becomes imaginary beyond a certain distance. Actually these interactions can be used to approach Yukawa. And of course dimensional analysis let us to implement yukawa directly, too, but this is very ad-hoc (while q=1/2, p=2 fits very well with $$P=\sqrt{E^2-M^2}$$ using the procedure some messages above).
If we consider masses in the interacting particles, we should by symmetry to keep them either as a sum m+m', and then it is as the previous case, or as a product, mm', getting two additional possibilities with adimensional coupling:
$$f=K mm'$$, $$f={K \over m m' x^4}$$
The second one shows a dependence similar to Fermi weak force (I supposse? Compare its potential with a delta function in three dimensions) except for masses. As for the first one, a constant force happens for instance in confinement of quarks, ie color string, but no mass dependence happens in this case, neither in fermi force. Still, Nature likes to keep things in order, and the string tension of charmonium is of the order of the product of masses of both quarks.
There are two cases where the coupling constant can be home of a scale. Gravity (ie geometry) and spontaneus symmetry breaking. In the second case, if a theory is the result of a very low energy approximation to a Yukawian theory, the scale of this theory appears in the coupling constant, for instance $$K_F=k/M_W^2$$ relates Fermi theory with yukawian GSW model of electroweak interactions. In the first case the coupling carries a unit of area, Planck area. In both cases the resulting theory is not renormalizable.

 

[arivero]

The following excerpt is being circulated to s.p.s, but it is also relevant to this thread so I am posting it here for completeness. Veteran PF readers will notice the similarity with arguments in the "Kepler Length" thread,
https://www.physicsforums.com/showthread.php?t=14007

===============================
Let me sketch a couple of classical arguments that I think can be
argued to ask space time dimension to compactify to D<5.

Consider a bound gravitational orbit around a mass m. Kepler third law
for generic dimension is T^2=R^(D-1). For D=4 this is the usual law.
And if instead of the radious of the orbit we consider the total area
of the orbit, we can also write it as T^2=A^((D-1)/2). Here we see
that something special happens at D=5: the period in this space time
depends linearly of the area.

If we consider in detail the area sweept by a particle as time goes,
we appreciate that D<5 and D>5 are different issues: The area goes

$$2 A(t) = G^{1/2} m^{1/2} R^{(5-D)/2} t$$

Thus for D>5, for a given time interval the area decreases with radius
of the orbit, while for D<5, areas increase with the radius of the
orbit.
It should be a surprise if this change in the trend of gravitational
bound states were not noticed in the spectrum of any theory having
classical Newtonian gravity as a limit. Thus D<5 should be a natural
point for compatitification.

In favour of LQG/discrete area theories, it is worth to notice what
happens if we ask A(t) to be a integer multiple of Planck area when t
is Planck time. Except for D=5, this requisite fixes a quantification
of the radius, but only for D=4 the gravitational constant G cancels
out, leaving just h, c, and m. This is another signal of the
privileged status of our current space time, or at least of its
delicate relationship with the concepts of time and area.

Alejandro Rivero

 

[arivero]

Thus for D>5, for a given time interval the area decreases with radius of the orbit, while for D<5, areas increase with the radius of the
orbit.

Perhaps one would add that the D=5 case will surely tip over one or another side when we use general relativity instead of the Newtonian aproximation.

 

[arivero]

Er... it seems that the edit button has disappeared today, so I am posting a new message simply to tell that I have abstracted the previous argument in a very short note, http://dftuz.unizar.es/~rivero/research/simple.pdf

 

[arivero]

Lets try to add some masses but still insisting in scale-less coupling constant. First, a massive propagator "M" could be used just to reduce or to increase one degree of force. So we have the possibilities
$$f= K M/x$$, $$f=K 1/Mx^3$$.
But in such cases the limit $$M \to 0$$does not recover the previous forces (thus should we need to colapse one spatial dimension in the limit if we ask for consistency?). Still it is possible to use M just to cancel the distance in other way:
$$f={K \over x^2} (1-(M x)^p)^q$$
If q=1/2 then we have an interaction that becomes imaginary beyond a certain distance. Actually these interactions can be used to approach Yukawa. And of course dimensional analysis let us to implement yukawa directly, too, but this is very ad-hoc (while q=1/2, p=2 fits very well with $$P=\sqrt{E^2-M^2}$$ using the procedure some messages above).

I was thinking on revisiting this argument. Can anybody point to the force equation above (or the corresponding potential) in some textbook? It is a sort of well but not exactly; the integration of

$$f={K \over x^2} \sqrt {1-(M x)^2}$$

gives (almost from memory, so please check if you have some mathlab/mathematica/maple nearby) an arcsin (x M) well plus an sqrt( 1- Mx2) / x thing. So it seems it is not yukawian after all but confining.