Thread: Higgs mechanism / is mass a transient phenomenon?

1. Hi,

From what I understand about the Higgs mechanism, particles acquire their mass by absorbing ziggs bosons. So my question is, does the particle have a mass only when it has absorbed the ziggs boson? As soon as it reemits it, it loses its mass.

In other words, mass is transient, and the particle actually alternates between mass/ no mass and from going at the speed of light/ being not at the speed of light. Because of that, the motion of the particle is jerky.

In any case, at the microscopic level, it is impossible to say that a particle will absorb and reemit a ziggs at an extremely precise time. The absorbtion/reemission will be cahotic and will follow a probability law. So the particle should have a jerky motion, shouldn't it?

Also when a particle is accelerated, the observer at rest will see the particle absorb/emit more ziggs. It's like say when an electron is accelerated, the observer at rest will see it emit photons and Z bosons. It is only an observer effect due to the relative speed. If it works the same way for the mass, the more an object is accelerated, the more mass it gets ( for an observer at rest ), so the notion of relativistic mass does make sense. I know that physicists usually don't like to use the notion of relativistic mass, but doesn't the Higgs mechanism legitimate that notion?

Nic.

2.

3. No. First, its Higgs. And no, higgs bosons themselves have nothing to do with actual mass of other particles. Other particles gain mass through interaction terms in lagrangian with higgs field after symmetry breaking. If you know what lagrangian is at least on classical level I can explain further and show on some basic model however if you dont it probably wouldnt give you anything and would just serve to confuse you more. Your choice :-)

4. I have seen the term ziggs in one of Leonard Susskind's lectures which was a introduction to the Higgs boson.

Here it is at 48:00

I searched the net to find out what really he meant by that and here is what I found:

Ziggs boson

The Ziggs particle(s) Susskind is referring too actually has several names in the literature, and they aren't always consistent (they are sometimes labeled differently in different textbooks etc). What he is presenting is a simplified picture of a more complicated story, although he makes this distinction to emphasize that there is in fact such a story.

It is important to note that this is not the Higgs boson or the Z boson.

Basically what he is referring to are the physical excitations of the Higgs vacuum condensate. In the standard model, this is actually represented by a 4 component field of 'Higgses'.

I will call these guys H+, H-, H0, h. This field interacts (in a complicated way) with 4 different massless gauge bosons that I will call W1, W2, W3 and B, where certain linear combinations of the W's mix into states that then get eaten by the Goldstone bosons in order to finally create the massive W+, W-, Z (and residual massless photon).

The Higgs boson (the h) is basically the residual excitation of the radial excitation of this full potential.

Flip Tanedo does a good job of explaining this in a series of posts.

http://www.quantumdiaries.org/2011/1...etry-breaking/
Its usage of the term 'ziggs' is a simplification.

I have seen the Lagrangian used several times but I am not really accostumed to it. I am gonna try to read on it to make sure I understand it so that you don't waste time giving a detailed explanation.

I am going to look into the Lagrangian, but actually my question was quite simple. Does an electron for instance always have a mass, or does it alternate between having a mass, having no mass, every fraction of a second ( by absorbing particles from the condensate/ emitting particles into the condensate )? In other words is the mass an intermittent phenomenon?

5. Ah, ok. No, electron is always massive. It doesn´t jump between massive and massless modes. You may however encounter something called zitterbewegung or something like that from which you may wrongly deduce that electron is really massless but this is quite different issue with solutions of dirac equation. It has nothing to do with higgs mechanism. Just so you don´t get confused by that if you encounter it somewhere.

edit: thanks for that ziggs reference btw, I never heard that name before

6. In the video Susskind says that the mass is proportional to the frequency of absorbtion of the ziggs ( let's use the term ziggs even though it is not the real scientific term to keep it simple ).

I still don't understand why absorbing the ziggs would prevent a particle from going at the speed of light. Why couldn't a massless particle for instance interact with a condensate and stay at the speed of light?

Has the Higgs theory been unified with the Newton laws of motion, and special relativity?

For instance, since the momentum is conserved ( not talking about special relativity ) , if a particle were to have 10 times more mass, that would mean it would absorb/emits 10 times more ziggs , so its speed would be 10 times less. Why would the product of the frequency at which the particle absorbs ziggs with the velocity be constant? That is kind of bizarre.

Also, why is the rest energy of a particle proportionnal to the frequency at which the particle absorbs ziggs and the speed of light squared? Are those sorts of things explained by the Higgs theory?

7. OK, I will have to go into a bit of detail. I think that Susskind while attempting to simplify the problem is actually making it more complicated.

1.Lagrangian
So a bit of introduction. Lagrangian L is a function which completely describes dynamics of a system. This may be a simple classical mechanical system (eg. pendulum) or even complex quantum problem withinfinite degrees of freedom (quantum fields). The trick is that using something called principle of least action one can get equations of motion directly from lagrangian. So for example instead of describing quantum electrodynamics by equation of motion for free electrons and photons and equation of motion for electron interaction with EM field you can just write Lagrangian wich contains all information.

For example:

is lagrangian of electromagnetic field. If you use principle of least action on this you will get

which are Maxwell equations.

2.Nambu-Goldstone model
Consider lagrangian looking like this

This is lagrangian of complex free (noninteracting) massless scalar field. Its equations of motion are wave equations

.

But what happens if this field interacts with itself? What if lagrangian woud look like this?

The second term is sort of mass term for scalar field with "wrong" sing and last term is quartic self interaction (two particles collide and change their momenta). So far this is just model and interaction terms serve as sort of effective potential. You could write this lagrangian schematicaly as

Using standard calculus techniques you can investigate properties of this potential. You will easily find out that in 3D graph with axis being andand V this is precisely the mexican hat potential. The potential in fact depends not on two quantities but single one which can be defined as

therefore the potential is rotationaly symmetrical and depends only on "distance" from the middle. The true minimum here is in fact not a point but a circle for which . This is vacuum value of this scalar field. Therefore vacum expectation value of this scalar field is nonzero. One sould perhaps use the deviation from his vacuum value as true dynamical value rather that and . We can recast lagrangian in polar coordinates ( and which is phase/angle) first. This gets

Now we will define new dynamical variable (H) as deviation from vacuum expectation value of scalar field like this. Let

Now writing lagrangian in this new for with H and as dynamical variables one gets (up to factor of 2 maybe)

The third term is mass term because if we use principle of least action with respect to H we get

which is equation of motion for massive scalar field with mass being . remains massless (this is goldstone boson) the remaining interaction term are of no interest to us.

There is nowhere any nontrivial assumtion this is just mathematical recast of original lagrangian. What we found out is that lagrangian

instead of describing two scalar charged fields with self interactions it describes one massive and one massless real scalar field + ineractions between those.

3.Abelian Higgs model
Now what happens if we try to transform the lagrangian in new coordinates that will be different by local phase? These are called gauge transformations.

The derivations in lagrangian also apply to therefore this reults in additional terms which means this lagrangian is not symmetric under these transformations. However it can be shown that this symmetry is kind of essential as it among others preserve quantities like electric charge which we know are conserved. Therefore we would like to repair our lagrangian to be gauge invariant (symmetric under these transformations). For example we know that electromagnetic fourpotential has some gauge freedom. This means that in particular fourpotential A and four potential yield the same lagrangian and therefore same Maxwell equations. We can use this like this by coupling gauge field (EM for example) to this scalar field like this

this results in fact that additional terms invoked by derivations of can be eaten by redefinition of but this has no dire consequences because A has certain gauge freedom. Therefore this whole lagrangian is now gauge invariant as it should be. We can now write whole thing in representation of H and as we have done earlier and recast it as

Second term is mass term for field A. Therefore what really happens is that A becomes massive gauge field (spin 1), H is massive scalar field,is massless scalar field like before. Dirty little secret is that mass of gauge field is given only by strength of coupling between scalar and gauge field and vacuum expectation value of field. That in turn depends on parameters of selfinteracting potential V. Not a single thing here is dynamical variable those are all constants. The true dynamical variable here is H which is of course Higgs field but as you can see this does not enter the mass term only in other interaction terms. This is of course only a model of coupling between gauge field and scalar field with this type of self interactions (so called Abelian higgs model). I am not saying anything like that photon is massive and such this is only mathematics. Reality is a bit different but the trick or mechanism through which mass terms for massive particles are introduced are same.

To answer your remaining question, yes, this is compatible with SR, in fact SR is one of main ingredients. What I think Susskind means by ziggs is interaction written with original \rho. That is clearly stupid approach because instead of massive gauge fields + Higgs one has massless gauge field which interacts with condensate of \rho field which has nonzero expectation value in vacuum. In principle both aproaches are valid but standard aproach of using H as true dynamical variable (excitations around vacuum) is clearly much more clear and simple but in the end this all ends up being a choice of variables.

I hope this helps you. You dont need to understand everything here. The important thing conceptualy for question "how Higgs works" is fact that you can redefine variable of scalar field as excitation around vacuum expectation value and basic understanding of gauge invariance which leads to interaction terms between scalar field and gauge field which in turn gives mass to gauge field. The case with electrons is a bit more complicated but theory is simple. Just adding of Yukawa interaction terms imedietaly gives mass terms for electrons, muons etc. + neutrinos if you want. I hope I didnt do to many mistakes in there

8. Thanks a lot Gere for this detailed explanation.

I will look at it carefully as soon as I can.

9. I watchted Susskind lecture about the Lagrangian and principle of least action, I get it now.

I am not really used to using tensors unfortunately. I follow a bit what you say in your points 1 and 2, but I have not yet really looked in detail at point 3.

I understand that it is because of the term V( phi) that the symmetry is broken.

The third term is mass term because if we use principle of least action with respect to H we get
...
I don't understand why.

Sorry if this is basic, but what is the notation of the partial derivative with the upper indice?

I am going to continue my investigations.

10. Originally Posted by Nic321
I understand that it is because of the term V( phi) that the symmetry is broken.
Exactly. This leads to degeneracy of vacuum state and nonzero expectation value.

Originally Posted by Nic321
The third term is mass term because if we use principle of least action with respect to H we get
...
I don't understand why.

You have lagrangian dependent on several variables, notably gauge field A and its derivatives, scalar field H and its derivatives and scalar field pi and its derivatives. If you do variation only in variables H and its derivatives you will get equation of motion only for field H. If you do variation in variables A and its derivatives you will get only Maxwell equations and so on. The equation of motion you get for H is called Klein-Gordon equation which is basic relativistic equation for massive scalar field.

Originally Posted by Nic321
Sorry if this is basic, but what is the notation of the partial derivative with the upper indice?
That is just special relativistic notation which differs between covariant fourvectors and contravariant fourvectors. Those are connected by metric tensor which has nonzero only diagonal components . Depends on convention it can be other way around with g00=-1 (that is convention usually used by relativists).

Therefore . Einstein summation convention is implicitly used (summing over same indiced on top and bottom). Therefore in this convention dAlembert wave operator (speed of light is 1).

For better introduction to relativistic conventions regarding tensors I would recommend http://www.thescienceforum.com/physi...ty-primer.html point 4 of Marcus GR primer.

11. Originally Posted by Gere
Originally Posted by Nic321
I understand that it is because of the term V( phi) that the symmetry is broken.
Exactly. This leads to degeneracy of vacuum state and nonzero expectation value.
Ok.

Originally Posted by Nic321
The third term is mass term because if we use principle of least action with respect to H we get
...
I don't understand why.

You have lagrangian dependent on several variables, notably gauge field A and its derivatives, scalar field H and its derivatives and scalar field pi and its derivatives. If you do variation only in variables H and its derivatives you will get equation of motion only for field H. If you do variation in variables A and its derivatives you will get only Maxwell equations and so on. The equation of motion you get for H is called Klein-Gordon equation which is basic relativistic equation for massive scalar field.
So you get the laws of motion for special relativity? which would be (dmu dmu - mu^2/2 ) H=0

Originally Posted by Nic321
Sorry if this is basic, but what is the notation of the partial derivative with the upper indice?
That is just special relativistic notation which differs between covariant fourvectors and contravariant fourvectors. Those are connected by metric tensor which has nonzero only diagonal components . Depends on convention it can be other way around with g00=-1 (that is convention usually used by relativists).

Therefore . Einstein summation convention is implicitly used (summing over same indiced on top and bottom). Therefore in this convention dAlembert wave operator (speed of light is 1).

For better introduction to relativistic conventions regarding tensors I would recommend http://www.thescienceforum.com/physi...ty-primer.html point 4 of Marcus GR primer.

I understand better up to your Abelian Higgs model paragraph. I have to search more on gauge invariance and the electromagnetic fourpotential to be able to understand ( maybe not completely but to have a general understanding ).

From what I understand, with this tool of lagrangian/principle of least action, you work backward from the laws of physics to build a Lagrangian, which gives you the fields and the interactions. Is that the goal of all these manipulations?

12. Originally Posted by nic321

Originally Posted by nic321
the third term is mass term because if we use principle of least action with respect to h we get
...
i don't understand why.

you have lagrangian dependent on several variables, notably gauge field a and its derivatives, scalar field h and its derivatives and scalar field pi and its derivatives. if you do variation only in variables h and its derivatives you will get equation of motion only for field h. if you do variation in variables a and its derivatives you will get only maxwell equations and so on. The equation of motion you get for h is called klein-gordon equation which is basic relativistic equation for massive scalar field.
so you get the laws of motion for special relativity? Which would be (dmu dmu - mu^2/2 ) h=0

klein-gordon equation is quantum mechanical equation that satisfies relativity.

Originally Posted by nic321

i understand better up to your abelian higgs model paragraph. I have to search more on gauge invariance and the electromagnetic fourpotential to be able to understand ( maybe not completely but to have a general understanding ).
the main idea is to use gauge freedom of electromagnetic potentials to "absorb" additional terms that arise because of derivations of . That preserves invariance of lagrangian and coresponding equations of motion to these types of transformations. These gauge fields you have to "invoke" are all spin 1 bosons eq. photons, W+-, Z, gluons.

Originally Posted by nic321

from what i understand, with this tool of lagrangian/principle of least action, you work backward from the laws of physics to build a lagrangian, which gives you the fields and the interactions. Is that the goal of all these manipulations?
[/quote]

Lagrangian is only one possible aproach to field theories. You can work from hamiltonian too but that path is much more complicated and I dont understand it very well. Some textbooks use principle of least action as sort of starting point of all physical theories. For example you can derive Newton equations from it and much more. However the use of PoLA hinges on knowledge of full lagrangian of the system. In my mind the principle of least action should not be looked at as some fundamental physical law rather a mathematical tool that along with knowledge of lagrangian leads to one possible description of system.

Id like to ask someone with deeper knowledge of relativistic QM: Susskind in that video talks about free electron constantly switching between its helicity states. Is that true? Doesnt seem right to me.

13. klein-gordon equation is quantum mechanical equation that satisfies relativity.
Is there only one such equation that sarisfies relativity? Is there like one per type of spin or whatnot?

the main idea is to use gauge freedom of electromagnetic potentials to "absorb" additional terms that arise because of derivations of . That preserves invariance of lagrangian and coresponding equations of motion to these types of transformations. These gauge fields you have to "invoke" are all spin 1 bosons eq. photons, W+-, Z, gluons.
Does this mean that the weak and strong forces are seen as electromagnetic phenomena? Is this how electromagnetism has been unified with the weak and stong forces or doesn't have anything to do with it?

Lagrangian is only one possible aproach to field theories. You can work from hamiltonian too but that path is much more complicated and I dont understand it very well. Some textbooks use principle of least action as sort of starting point of all physical theories. For example you can derive Newton equations from it and much more. However the use of PoLA hinges on knowledge of full lagrangian of the system. In my mind the principle of least action should not be looked at as some fundamental physical law rather a mathematical tool that along with knowledge of lagrangian leads to one possible description of system.
The Lagrangian seems to be a pretty effective tool. Just out of curiosity, is the fact that the Lagrangian can be used to describe physical laws linked to conservation law of a certain quantity, say energy for instance? Do we know why it works or do we just trust that physical laws probably follow a principle of least action?

14. Originally Posted by Nic321
klein-gordon equation is quantum mechanical equation that satisfies relativity.
Is there only one such equation that sarisfies relativity? Is there like one per type of spin or whatnot?
There is one per spin type. Klein-Gordon is for scalars, Dirac is for spin 1/2 fermions, Proca equation (sort of Maxwell equations) is for spin 1, spin 3/2 has Rarita-Schwinger and 2 has Pauli-Fierz equation. Last two have some interpretation problems though as I understand it.

Originally Posted by Nic321
the main idea is to use gauge freedom of electromagnetic potentials to "absorb" additional terms that arise because of derivations of . That preserves invariance of lagrangian and coresponding equations of motion to these types of transformations. These gauge fields you have to "invoke" are all spin 1 bosons eq. photons, W+-, Z, gluons.
Does this mean that the weak and strong forces are seen as electromagnetic phenomena? Is this how electromagnetism has been unified with the weak and stong forces or doesn't have anything to do with it?
Well in certain sense. The gauge freedom is general property of all these fields and they are introduced all in same manner as gauge fields. There are some fundamental distinctions though. As you might know the W and Z are massive thus short range forces + W are charged. Z is closest example of what you may call massive photon. Their properties are very similar. Gluons are massless but interact with themselves therefore their behaviour is quite different from photons. They are all different but come from same basic idea. You may have heard of Yang-Mills theory, this is exactly it.

Originally Posted by Nic321
Lagrangian is only one possible aproach to field theories. You can work from hamiltonian too but that path is much more complicated and I dont understand it very well. Some textbooks use principle of least action as sort of starting point of all physical theories. For example you can derive Newton equations from it and much more. However the use of PoLA hinges on knowledge of full lagrangian of the system. In my mind the principle of least action should not be looked at as some fundamental physical law rather a mathematical tool that along with knowledge of lagrangian leads to one possible description of system.
The Lagrangian seems to be a pretty effective tool. Just out of curiosity, is the fact that the Lagrangian can be used to describe physical laws linked to conservation law of a certain quantity, say energy for instance? Do we know why it works or do we just trust that physical laws probably follow a principle of least action?
Well we know why it works, thats just simple mathematics that always works. You need to know the lagrangian though. Conservation laws are connected with lagrange formulation by something called Noethers theorem (Emma Noether). As long as action (integral of lagrangian) is symmetric under certain continuous symmetry operation there exists conserved fourcurrent AND conserved quantity. These symmetry operations may be spacetime symmetries (Poincare symmetries) or intristic symmetries like gauge symmetry. This theorem therefore says that things like energy, momentum, angular momentum, position of center of mass, electric charge etc. are all conserved.

15. There is one per spin type. Klein-Gordon is for scalars, Dirac is for spin 1/2 fermions, Proca equation (sort of Maxwell equations) is for spin 1, spin 3/2 has Rarita-Schwinger and 2 has Pauli-Fierz equation. Last two have some interpretation problems though as I understand it.
It take one per spin because of the conservation of spin I guess ? spin 2 is for the graviton, isn't it, is that for general relativity?

Well in certain sense. The gauge freedom is general property of all these fields and they are introduced all in same manner as gauge fields. There are some fundamental distinctions though. As you might know the W and Z are massive thus short range forces + W are charged. Z is closest example of what you may call massive photon. Their properties are very similar. Gluons are massless but interact with themselves therefore their behaviour is quite different from photons. They are all different but come from same basic idea. You may have heard of Yang-Mills theory, this is exactly it.
I have read a bit about the Yang-Mills theory, and it is non abelian, whereas the Higgs theory is abelian from what you said before.

So from what I understand some self interactions are abelian and some are not, such as the Higgs coupling with the other fields. I don't quite follow what change it makes if the coupling is abelian or not?

Well we know why it works, thats just simple mathematics that always works. You need to know the lagrangian though. Conservation laws are connected with lagrange formulation by something called Noethers theorem (Emma Noether). As long as action (integral of lagrangian) is symmetric under certain continuous symmetry operation there exists conserved fourcurrent AND conserved quantity. These symmetry operations may be spacetime symmetries (Poincare symmetries) or intristic symmetries like gauge symmetry. This theorem therefore says that things like energy, momentum, angular momentum, position of center of mass, electric charge etc. are all conserved.
Ok, the Lagrangian is a practical tool to easily "encode" laws of physics that must conserve certain quantities under certain symmetries. This is technical but at least I understand the general idea.

16. Nice and clear explanations, Gere - well done

17. Originally Posted by Nic321
There is one per spin type. Klein-Gordon is for scalars, Dirac is for spin 1/2 fermions, Proca equation (sort of Maxwell equations) is for spin 1, spin 3/2 has Rarita-Schwinger and 2 has Pauli-Fierz equation. Last two have some interpretation problems though as I understand it.
It take one per spin because of the conservation of spin I guess ? spin 2 is for the graviton, isn't it, is that for general relativity?
Conservation of spin has nothing to do with that. Its just that particles with different spin behave differently. Spin 2 should be for quantized gravity but I know almost nothing about this subject.

Originally Posted by Nic321
Well in certain sense. The gauge freedom is general property of all these fields and they are introduced all in same manner as gauge fields. There are some fundamental distinctions though. As you might know the W and Z are massive thus short range forces + W are charged. Z is closest example of what you may call massive photon. Their properties are very similar. Gluons are massless but interact with themselves therefore their behaviour is quite different from photons. They are all different but come from same basic idea. You may have heard of Yang-Mills theory, this is exactly it.
I have read a bit about the Yang-Mills theory, and it is non abelian, whereas the Higgs theory is abelian from what you said before.

So from what I understand some self interactions are abelian and some are not, such as the Higgs coupling with the other fields. I don't quite follow what change it makes if the coupling is abelian or not?
The model I showed is just simple toy model how could higgs give mass to gauge field. In standard model Higgs is coupled to nonabelian Yang-Mills fields + Yukawa interaction between Higgs dublet and lefthanded isospin dublet + righthanded singlets to get mass to leptons. Quark sector is even more complicated.

The fact that some theories are nonabelian only results in additional interaction terms between those gauge fields (eq between Z,W and photon). For example if chromodynamics was somehow abelian there would be no interaction between gluons but as we know SU(3) is nonabelian group therefore the 8 generators (gluons) do couple between themselves. I could show this but you would need much more knowledge in both mathematics and qft for it to give you something.

Originally Posted by Markus Hanke
Nice and clear explanations, Gere - well done
Wow, thank you Marcus :-)

18. Thanks a lot Gere for all these explanations. I think going further would not be really useful because of my lack of bases about the topics you are refering to. I am already at the limit of what i can understand without significantly more research.

In any case it was very interesting!

19. Originally Posted by Gere
Wow, thank you Marcus :-)
No problem
Could I ask what your comments with regards to this would be : Path Integrals and Topology

20. Thanks Nic

Originally Posted by Markus Hanke
Originally Posted by Gere
Wow, thank you Marcus :-)
No problem
Could I ask what your comments with regards to this would be : Path Integrals and Topology
I´m sorry Markus but I am unable to help with path integrals. When I tried to learn path integral formulation (from book in my own native language) I got lost at something like page two when they introduced Wiener measure :-) My friend who actually attended course of path integral told me that it was most difficult lecture he attended (and he attended some really juicy ones like chiral symmery in chromodynamics, effective lagrangians in qft, beyond standard model etc.) so I am hesiteting whether to even try :-D Intuitively I would say that integral should be diffeo. invariant because action is invariant but I won´t even try to guess what would happen after some change in topology.

21. Originally Posted by Gere
I´m sorry Markus but I am unable to help with path integrals. When I tried to learn path integral formulation (from book in my own native language) I got lost at something like page two when they introduced Wiener measure :-) My friend who actually attended course of path integral told me that it was most difficult lecture he attended (and he attended some really juicy ones like chiral symmery in chromodynamics, effective lagrangians in qft, beyond standard model etc.) so I am hesiteting whether to even try :-D Intuitively I would say that integral should be diffeo. invariant because action is invariant but I won´t even try to guess what would happen after some change in topology.
No problem, thanks anyway. I am fairly sure it is diffeomorphism invariant ( since the paths a system can take in configuration space cannot be intrinsically dependent on space-time coordinates ), just wanted to see how to proof this mathematically. As for changes in topology, this question has been on my mind for some time, and somehow it seems like no one is able to answer it, and I couldn't really find any info about it online either. I posted this question on PhysicsForums, but they judged it to be too speculative, and the thread was disallowed. A real pity. My original impetus was Wheeler's geometrodynamics, i.e. attempting to model elementary particles not as excitations of quantum fields, but rather as geometric and topological features of space-time itself; as such I was interested to see what would happen if, for a given quantum system, we change the underlying space-time from a singly connected one to a multiply connected one ( e.g. a microscopic Einstein-Rosen bridge ), and so the question naturally arose whether it is possible to find a description of a given system that is independent not just of the coordinate basis, but also of the topology of space-time itself. In other words - if diffeomorphism invariance lies at the heart of GR's curved space-time, then what kind of physics would topological invariance give rise to ?

Unfortunately my mathematical knowledge is nowhere near sufficient to answer this question myself.

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